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An indicator formula for a semi-simple Hopf algebra

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An indicator formula for a semi-simple Hopf algebra

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Content AN INDICATOR FORMULA FOR A SEMI-SIMPLE HOPF ALGEBRA by Kayla Orlinsky A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) May 2023 Copyright 2023 Kayla Orlinsky Dedicated to those who supported me, especially the four-legged. ii Acknowledgments It was directly in the middle of my program that the Covid-19 pandemic hit. During the tumultuous year that followed lock-down, one of the very few people I continued to speak to in person was my advisor. I would like to thank my advisor Dr. Susan Montgomery for graciously opening her home to me so that we could sit outside on her patio–six feet apart wearing masks–and discuss math, life, and current events. From the food (pizza, salad, homemade cookies, and BBQ)tothefunwithSusan’scatSophie, theweeklydrivesupPCHandviewsoftheocean and Susan’s eclectic garden kept me sane during lock-down. I would also like to thank my friends who comprised of fellow graduate students, former students from my undergraduate institution, and friends from childhood who remained present in my life during some of my greatest moments of uncertainty. Finally, I would like to thank my family for supporting me when major life decisions drewmeawayfromgraduateschoolandleftmyfutureplansuncertain. Itwasduringtimes of variability that I learned the most about myself and ultimately the decision to reapply to graduate school would not have happened without your encouragement. iii Table of Contents Dedication ............................................................ ii Acknowledgments....................................................... iii List of Tables .......................................................... vi List of Figures ......................................................... vii Abstract .............................................................. ix Chapter 1: Chapter Introduction .......................................... 1 1 Representation Theoretic Motivation for Hopf Algebras . . . . . . . . . . . 1 2 Frobenius-Schur Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Significance of the Indicator for Hopf Algebras . . . . . . . . . . . . . . . . 4 4 Motivating the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Chapter Group Results......................................... 9 1 Group Action Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 3: Chapter Hopf Algebras ......................................... 13 1 Hopf and Representation Results . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 4: Chapter Understanding the Action................................ 20 1 Reinterpreting the Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Reinterpreting Orbits and Stabilizers . . . . . . . . . . . . . . . . . . . . . . 20 3 Uncovering New Permutations Properties . . . . . . . . . . . . . . . . . . . 25 Chapter 5: Chapter Indicators of J n ........................................ 29 1 A Revised Indicator Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 6: Chapter Prerequisite Counting Recursions.......................... 36 1 Structure of Permutations with Given Stabilizer. . . . . . . . . . . . . . . . 36 2 Counting Involutions with Certain Properties . . . . . . . . . . . . . . . . . 40 3 Prerequisites for Counting Indicators . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 7: Chapter Counting ............................................. 57 1 Counting the Number of odd dimensional Irreps with Indicator +1 . . . . . 57 2 The Case t=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter 8: Chapter Limits ............................................... 67 1 Limiting Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Chapter A: Chapter Examples and Python Coding............................ 74 1 Discussing the Python Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2 Using Proposition 6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3 Using Proposition 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Using Proposition 6.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Using Proposition 6.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6 Using Proposition 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7 Using Theorem 7.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 iv 8 Using theorem 7.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 9 Counting Irreps with Indicator +1 Directly . . . . . . . . . . . . . . . . . . 95 10 Counting Irreps with Indicator− 1 Directly . . . . . . . . . . . . . . . . . . 98 Bibliography........................................................... 100 v List of Tables 6.1 A table of sets and values which are used heavily in the following state- ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.1 A table summarizing some results which are now obtainable through a counting formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 A.10.1 Table showing total number of irreps of J n of dimension t with indicator − 1. This value was computed directly in Python. Note how t was kept small to make computation possible. . . . . . . . . . . . . . . . . . . . 98 vi List of Figures A.1.1 Excerpt from python code which calculates the orbit of a given permu- tation for fixed n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.2.1 A flow chart illustrating the process of Proposition 6.1.1. This will con- struct all permutations which are stabilized by a t . To get the exact set of permutations which have stabilizer ⟨a t ⟩, this process must be done recursively so that all x which are stabilized by a s for some s|t can be removed from the set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.2.2 Graph of M n/t : number of permutations x ∈ S n− 1 with stabilizer ⟨a t ⟩. Notethenon-linearscalingofthey-axiswhichwasimplementedtoensure the graph could be viewed for large n . . . . . . . . . . . . . . . . . . . 78 A.3.1 A flow chart illustrating the process of Proposition 6.2.1. This will con- struct all involutions which are stabilized by a t . To get the exact set of involutions which have stabilizer ⟨a t ⟩, this process must be done recur- sivelysothatallxwhicharestabilizedbya s forsomes|tcanberemoved from the set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.3.2 Graph of T n/t : number of involutions x∈S n− 1 with stabilizer⟨a t ⟩. Note the nonlinear scaling of the y-axis. . . . . . . . . . . . . . . . . . . . . . 81 A.4.1 Graph of R n/t,r : the number of involutions in S n− 1 with stabilier ⟨a t ⟩ and r fixed points. Note the non-linear scaling of the y-axis. . . . . . . 83 A.5.1 Graph of O n/t,r : the number of orbits of length t which contain r invo- lutions. Note the non-linear scaling of the y-axis. . . . . . . . . . . . . . 85 A.7.1 Graph of I (+1) n/t showing the number of irreps of J n of dim t (for odd t) with indicator +1. Note the nonlinear scaling of the y-axis. In order to emphasize the significance of the value n/t, a scatter plot was chosen instead of a line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 A.8.1 A flow chart illustrating the process of Theorem 7.2.2. The red entries indicate the construction of involutions x and the process of determining which indicators are +1 and which are 0. The green entries indicate the construction of x such thatO x ={x,x − 1 } (so namely, x̸=x − 1 ) and the process of determining which indicators are +1,− 1,0. The rectangles illustrate that F x = F if j = 2u + 1 mod n 2 , the diamonds illustrate intermediate computation, and the circles illustrate the final conclusion of the indicator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.8.2 Graph of I (0) n/2 showing the number of 2-dim irreps of J n with indicator 0. 93 vii A.8.3 Graph of I (+1) n/2 showing the number of 2-dim irreps of J n with indicator +1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A.8.4 Graph of I (− 1) n/2 showing the number of 2-dim irreps of J n with indicator − 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A.8.5 Plot showing all three of Appendix 8, Appendix 8, and Appendix 8 on a single graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A.8.6 Plot showing Figures Appendix 8 and Appendix 8 on a single graph. . . 95 A.9.1 Graph showing I (+1) n/t the number of irreps of J n of dimension t with indicator +1. Notice that here, when t is even, our formula is an under- estimate of the actual value. Note the nonlinear scaling of the y-axis. . 95 A.9.2 Graph showing a close up of Appendix 9 which highlights the underes- timate of our formula whenever t is even. Note the nonlinear scaling of the y-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 A.9.3 Graph showing the difference between direct computation and our for- mula for irreps of J n of dimension t (for even t) with indicator +1 . . . 96 A.9.4 Graph showing percent of 4-dimensional irreps of J n with indicator +1 to total number of 4-dimensional irreps of J n . . . . . . . . . . . . . . . . 97 A.10.1 Graph showing number of irreps of J n of dimension t with indicator− 1. This value was computed directly in Python. . . . . . . . . . . . . . . . 98 A.10.2 Graph showing percent of 4-dimensional irreps of J n with indicator − 1 to total number of 4-dimensional irreps of J n . . . . . . . . . . . . . . . 99 viii Abstract The semisimple bismash product Hopf algebra J n =k S n− 1 #kC n for an algebraically closed field k is constructed using the matched pair actions of C n and S n− 1 on each other. In this work, we reinterpret these actions and use an understanding of the involutions of S n− 1 to derive anew Froebnius-Schurindicator formula forirreps of J n and showthat for n odd, all indicators of J n are nonnegative. We also derive a variety of counting formulas including Theorem 7.2.2 which fully describes the indicators of all 2-dimensional irreps of J n and Theorem 7.1.2 which fully describes the indicators of all odd-dimensional irreps of J n and use these formulas to show that nonzero indicators become rare for large n. ix Chapter 1 Introduction Foundational understanding of groups, group actions, tensor products, and group represen- tation theory is helpful and perhaps necessary to understanding this work. Throughout this work, all groups will be taken to be finite and all fields will be alge- braically closed of characteristic 0. 1 Representation Theoretic Motivation for Hopf Algebras Given a group G, a representation of G is a pair (ρ,V ) where V is a vector space and ρ : G → GL(V) is a group homomorphism from G to the general linear group of V. Equivalently, if we consider the group algebra for some field k, kG= ( X x∈G α x x α x ∈k ) then we may view V as a kG-module where the action of kG on V is described by ρ via x· v := ρ (x)(v). Thus, it is common to refer to representations of groups only in terms of V or ρ . We will also use representation of G and kG-module interchangeably. Given two group representation, V and W, the tensor product V ⊗ W is again kG- module where the action is x· (v⊗ w)=(x· v)⊗ (x· w). This action is described using the k-linear map ∆: kG→kG⊗ kG x7→x⊗ x in the sense that if ρ V and ρ W are the group homomorphisms describing the action of G on V and W respectively, then ρ V⊗ W :=(ρ V ⊗ ρ W )◦ ∆. ThetrivialrepresentationiskthoughtofasakG-modulewheretheactionisx· α =α for α ∈k. This again can be described via the k-linear map ε:kG→k defined by ε(x)=1 k . 1 Finally, given a representation V of G, the dual V ∗ = {v ∗ : V → k|v ∗ is linear} is again a representation of G via the action (x· v ∗ )(w):=v ∗ (ρ (x − 1 )(w)). This action can be described using the k-linear map S :kG→kG x7→x − 1 in that ρ V ∗ (x)(v ∗ )(w):=v ∗ ((ρ V ◦ S)(x)(w)). Note that various compatibility conditions between ∆ ,ε,S and the algebra structure of kG are necessary. For example, we want k⊗ V ∼ = V ∼ = V ⊗ k as representations of G which implies a relationship between ∆ (which describes the action of G on a tensor product)andε(whichdescribestheactionof Gonk). Thatis, werequirem◦ (Id⊗ ε)◦ ∆= m◦ (ε⊗ Id)◦ ∆=Id where m is the multiplication of kG and Id is the identity map on kG. More generally, given an algebra H, if we want to be able to describe its representations and maintain that those representations can be tensored, dualized, and that the trivial representation exists, we must be able to equip H with the maps ∆ ,ε,S in a way which is compatible with the algebra structure of H. An algebra with such a structure is called a Hopf algebra. More generally, knowing certain properties of the category of modules of an algebra forces additional structure on that algebra and vice versa via Tannaka Duality. It must be noted here that Hopf algebras were first introduced by Heinz Hopf in the 1940s as a tool for understanding cohomology of compact Lie groups and their usefulness in representation theory was not discovered until decades later (see [C]). Hopf algebras now appear in many fields of mathematics including category theory, combinatorics, topology, geometry, algebra, functional analysis, quantum-field-theory, and condensed matter physics. 2 Frobenius-Schur Indicators LetGbeagroupand(ρ,V )bearepresentationofG. Thenthecharacterχ ofρ isdefinedby χ (x):=trace(ρ (x)). Recallthatsinceρ (x)∈Gl(V),foragivenchoiceofbasiswecanobtain amatrixrepresentationforρ (x). Notealsothatachangeofbasiswouldbedefined Aρ (x)A − 1 2 for some invertible matrix A over k. Since trace(ABC) = trace(CAB) = trace(BCA) for any square matrices A,B,C, we have trace(Aρ (x)A − 1 ) = trace(ρ (x)) and so the character is well defined and independent of choice of basis. A group representation V is called irreducible if it has no nontrivial G-invariant sub- spaces W. We will frequenlty refer to such representations as irreps. By G-invariant, we mean x· w ∈ W for all w ∈ W. An irreducible character is the character of an irreducible representation. The Frobenius-Schur [FS] indicator ν of a character χ is defined as ν (χ )= 1 |G| X x∈G χ (x 2 ). First discovered by its namesakes in 1906, the magic of the Frobenius-Schur indicator is that if χ is an irreducible character, ν (χ ) only takes the values 0, 1, or − 1. If k = C, these three values determine whether the character χ is complex-valued, real-valued and there exists a basis such that ρ (x) is real-valued for all x∈G, or real-valued with no basis for which ρ (x) is real-valued for all x∈G. In general, the values 0,1,− 1 of the indicator determine if the character admits a nondegenerate G-invariant bilinear form (ν (χ ) ̸= 0) and if so, if that form is symmetric (ν (χ ) = +1) or skew-symmetric (ν (χ ) =− 1). Recall, a bilinear form ⟨,⟩ : V ⊗ V → k is a functionsatisfying⟨av 1 +bv 2 ,w⟩=a⟨v 1 ,w⟩+b⟨v 2 ,w⟩and⟨v,cw 1 +dw 2 ⟩=c⟨v,w 1 ⟩+d⟨v,w 2 ⟩. A blinear form is called G-invariant if ⟨x· v,x· w⟩ = ⟨v,w⟩ for all x ∈ G, symmetric if ⟨w,v⟩=⟨v,w⟩ and skew-symmetric if⟨w,v⟩=−⟨ v,w⟩ for all v,w∈V. Inadditiontodeterminingtheexistenceofabilinearform, theindicatoralsogivessome insight into the structure of the group. For example, if Irr(G) is the set of all irreducible characters of G, then |{y∈G|y 2 =x}|= X χ ∈Irr(G) ν (χ )χ (x). More specifically, |{y∈G|y 2 =1}|= X χ ∈Irr(G) ν (χ )dim(χ ). 3 Although indicators are useful in group representation theory, the presence of countless othertoolsingrouptheorylimititsnecessity. Whereindicatorsreallyshineisinthetheory of Hopf algebras. 3 Significance of the Indicator for Hopf Algebras Movingforwardwewilluse ˆ V, ˆ ρ , and ˆ χ todescriberepresentationsofanalgebra H soasto distinguishthemfromtherepresentationsV,ρ, andχ ofagroup. Thiswillbecomenecessary later in the work when we induce algebra representations from group representations. Although[FGSV]extendedtheindicatortoKacalgebrasin1999,explicitlyconstructing an analogue of the Frobenius-Schur indicator for semisimple Hopf algebras was first done by Linchenko and Montgomery in 2000 [LM]. The key idea of [LM] came from the fact that a semisimple Hopf algebra H over an algebraically closed field admits a unique two- sided integral element Λ ∈ H. Namely, H has a unique (up to scalar multiple) element Λ satisfying hΛ = ε(h)Λ = Λ h for all h∈ H. Then, if ˆ χ is an irreducible character of H, plugging the second Sweedler power of Λ into ˆχ gives ν (ˆ χ )= ˆ χ ((m◦ ∆)(Λ))= ˆ χ X Λ 1 Λ 2 whichalsoonlyreturns0,1,or− 1. Furthermore,thesethreevaluesgivenbyν (ˆ χ )determine whethertheassociatedH-moduleadmitsanondegenerateH-invariantbilinearform(ν (ˆ χ )̸= 0) and if so, if that form is symmetric (ν (ˆ χ ) = +1) or skew-symmetric (ν (ˆ χ ) =− 1). It is therefore a perfect analogue of the Frobenius-Schur indicator for groups. Frobenius-SchurindicatorsonHopfalgebrasareofinterestforavarietyofreasons,three of which stand out. First, and most obviously, they determine if the associated H-module (representation) has an H-invariant, bilinear form and further if it is symmetric or skew- symmetric. In fact, it was shown by Linchenko and Montgomery [LM] that ν (ˆ χ )̸=0 if and only if the associated simple module (with character ˆ χ ) is self dual. Secondly, the indicator has been found to aid in the classification of finite dimensional Hopf algebras. This problem of describing all Hopf algebras of a certain dimension is in generalverydifficultandtedious. Forexample,in2000,Kashina[K]classifiedallsemisimple Hopf alebgras of dimension 16 and then in 2001, Kashina, Mason, and Montgomery [KMM] 4 showedthatthecomputationsofallHopfAlgebrasofdimension16couldbeimprovedusing the (second) indicator ν [KMM]. Also [KSZ1] used the indicator to put restrictions on the dimensionofaHopfalgebrabyprovingaspecialcaseofFrobenius’Theorem. Namely, they showed that if H has an even dimensional simple module, then the dimension of H itself must be even [KSZ1]. Thirdly, andmostgenerally, Frobenius-Schurindicatorscanbeadaptedtohavebroader applications to category theory. If H is a semisimple Hopf algebra over C, then we can define Rep( H), which is the category of finite dimensional representation of H. From our discussion in Section 1, Rep(H) is a pivotal fusion tensor category meaning that it is a category with finitely many isomorphism classes of simple objects (fusion) with a notion of dualization (pivotal) and tensor product (tensor). It was shown by Mason and Ng in 2005 [MN] that the Frobenius-Schur indicator is a categorical invariant. That is, if H and H ′ are two semisimple Hopf algebras over an algebraically closed field of characteristic 0 such thatRep(H)isequivalenttoRep(H ′ )ask-lineartensorcategories,then{ν (ˆ χ )} ˆ χ ∈Irr(H) and {ν (ˆ χ ′ )} ˆ χ ′ ∈Irr(H ′ ) are identical. Although we do not discuss them in this thesis, there are abundant applications if one were to expand out to higher indicators. That is, by computing the n th -Sweedler power of Λ (for n > 2 and plugging that into ˆ χ . For example, in the same paper where the idea is introduced, [KSZ2] prove a version of Cauchy’s Theorem for semisimple Hopf algebras. 4 Motivating the Research In this thesis, we expand our understanding of the indicators of the irreducible representa- tions of the semisimple Hopf algebra J n =C S n− 1 #CC n which is a bismash product of the two groups S n− 1 (the symmetric group on n− 1 letters) and C n (the cyclic group of order n). In general, the construction of Hopf algebras of the form k G #kF was first proposed by Kac and Palyutkin in 1960s and 1970s, however, their work was in C ∗ -algebras and was not well known to Hopf algebraists at the time. Thus, Takeuchi’s [Ta] definition in 1981 is more widely cited. 5 The significance of the Hopf algebra J n is that its construction arises from the groups S n− 1 and C n which form a factorization of the symmetric group S n = S n− 1 C n . Hopf algebras arising from symmetric groups are of interest to Hopf algebraists as there are many representation-theoretic results surrounding the symmetric group S n and some of these seem extendable to Hopf algebras. For example, in the 1940s, Alfred Young proved that the symmetric group S n is totally orthogonal, meaning for any χ ∈ Irr(S n ), ν (χ ) = +1. Then, in 1991, Scharf proved that all higher indicators of the symmetric group were nonnegative integers. In their groundbreaking paper of 2002, [KMM] proved that the irreps of the Drinfeld double Hopf algebra of S n , D(S n ) := k Sn #kS n are all +1–thus replicating Young’s result. However, the equivalent statement of Scharf–that all higher indicators are nonegative–is still open for the Drinfeld double of S n . Replicating the results of Young and Scharf for otherHopfalgebrasinvolvingthesymmetricgrouphasbeenanongoingprojectfordecades. It was shown by Masuoka [MA] that the Hopf algebra of interest in this paper, J n , is one of only two Hopf algebras that arise from the factorization of S n as a product of S n− 1 and C n . The other Hopf algebra being H n = (J n ) ∗ the dual of J n for which [JM] have already shown that all indicators of simple modules are +1. Similar work has been done by Timmer [Ti] in 2015 using different factorizations of the symmetric group S n = S n− r G and considering H = k G #kS n− r . Timmer [Ti] shows that the indicatoris nonnegative forirreducible characters ofhis algebras, aresult which we find to be false for J n for certain n. Some computations on the indicators of J n were also preformed by Schauenburg [S2]. We replicate their result [[S2] Theorem 4.7] (our Proposition 5.1.5 Item 4), although no efforts were made by [S2] to count irreps, which we successfully do in this work for all odd divisors of n. 5 Outline of the Thesis Specifically, in this thesis, we study the indicators of the simple modules (irreps) of a par- ticular semisimple Hopf algebra, specifically the bismash product J n =C S n− 1 #CC n where 6 S n is the symmetric group. We rely especially on the works of [MO] which served as a general resource on Hopf Algebras as well as [JM] which derived an indicator formula for the irreps of bismash product Hopf algebras. Additionally, a key result of [KMM] (Theo- rem 3.1.2) showed that there is a one-to-one correspondence between irreducible represen- tations (irreps) of J n and group representations of F x ⊂ C n where F x is the stabilizer of a permutation x∈S n− 1 under the matched pair action of C n on S n− 1 . Using this correspon- dence, we adapt the formulas of [JM] and find new ways to explicitly describe and count irreps of J n with certain indicators. Finally, we use code developed in Python to provide insight into negative indicators. The goal of this work was to fully describe all irreps of J n and state their second indicators ν (χ )=ν 2 (χ ). Whenn isodd, thiscanbedoneexplicitlyusinganunderstanding of involutions in the symmetric group S n− 1 and some computation. However, when n is even,moreworkisneededtocompletelydescribeallirrepsandtheirindicators. Specifically, when n is odd, we prove the indicator is nonnegative but for n ≥ 12 such that 4 divides n, there exists representations of J n with negative indicator which has complicated the problem. The organization of the paper is as follows. In Chapter 2, we discuss the group actions which determine the underlying structure of our Hopf algebra. This involves describing facorizable groups, which are groups L which contain two subgroups F,G such that F ∩G = 1 and L = GF = FG. Because we can write every element l ∈ L as a (unique) product xa ∈ GF = L with a ∈ F and x ∈ G, two natural group actions arise of these subgroups on each other which describe how to write xa uniquely as an element of FG. The factorizable group of interest to us will be the symmetric group of order n, S n which we will show can be written as a product of C n and S n− 1 ,whereC n hereisthecyclicgroupgeneratedbythestandardn-cyclea=(1 2 ··· n). We will discuss the orbits, stabilizers, and other useful sets and give some lemmas which we will need in order to prove our indicator counting formulas later on. In Chapter 3, we explicitly describe the Hopf algebra J n and summarize relevant results in[JM], whichincludeadescriptionoftheindicatorsof H n =(J n ) ∗ thedualofJ n . In[JM], a complete description of indicators is given for J p where p is prime, and also for H n for all n. While we have not been able to fully describe all indicators of J n for all n, in this 7 chapter we discuss the methods used for when p is prime and provide some useful results which apply for general n. In Chapter 4, we completely re-interpret the actions of C n on S n− 1 . We describe how viewing each permutation in the symmetric groups as a bijection on Z/nZ and then using some modular arithmetic, we uncover new relationships within the orbits and stabilizers of the group actions described in Chapter 3. For example, when we consider the action of C n = ⟨a⟩ on S n− 1 , we show that if a permutation x ∈ S n− 1 (viewed as a permutation in S n which fixes n) has stabilizer F x = ⟨a t ⟩, then x exhibits a number of remarkable propertiesincludingasortoflinearityin t. WeusethesepropertiestoderiveTheorem5.1.2 in Chapter 5 which is an improved indicator formula which drastically reduces–but does not eliminate–the computation requirement. Using this, we replicate a result of [S2] which statesthatforanynnotdivisibleby4,theindicatorofanyirrepofJ n isalwaysnonnegative. In Chapter 6, we cover majority of key results for this paper. Throughout this chapter, we use the relations discovered in Chapter 4 to fully construct (and therefore count) many sets of interest. We show that any permutation x∈ S n− 1 with stabilizer ⟨a t ⟩⊂ C n can in fact be constructed from some permutation in S t− 1 . In Chapter 7, we use the formulas from the previous chapter to show that if n is odd, then counting irreducible representation of J n is equivalent to counting involutions of S n− 1 with certain properties and, given an odd integer t, provide a formula for counting all t- dimensional irreps with indicator +1 (Theorem 7.1.2). Although we cannot describe all indicators of t-dimensional irreps of J n when t is even, we are able to fully describe the indicators of all 2-dimensional irreps of J n (Theorem 7.2.2). In Chapter 8, we discuss limiting behavior using the formulas we developed previously. We show that the ratio of nonzero irreps of dimension t (odd or t=2) to the total number of irreps of dimension t tends to 0 as n→∞ (Theorem 8.1.4 and Theorem 8.1.6). Finally, in Appendix A, we give examples and show how to use the formulas given to count orbits, involutions, and indicators of J n . 8 Chapter 2 Group Results 1 Group Action Results In this chapter, we describe the underlying group actions which make up the structure of the bismash product Hopf algebra J n . Throughout, L,F,G will be finite groups. Elements a,b∈F and x,y∈G and l∈L. Considera(finite)group LwithtwopropernontrivialsubgroupsF andGwhichintersect trivially and whose product is the entire group L. We call such a group factorizable and such objects have been studied by group theorists since the 1940s. Definition 2.1.1. A group L is factorizable if it has two subgroups F and G such that F ∩G=1 and L=GF =FG. Being able to write that L=GF and knowing that F∩G=1 implies that L=FG and it also implies that every element l∈L can be written uniquely as a product of something in F and something in G. Note we cannot say that L is a direct (or even semi-direct) product of F and G because nowhere are we assuming normality of either subgroup. Definition 2.1.2 ([Ta]). Let F and G be groups. Let◁:G× F →G and▷:G× F →F be group actions. Then (F,G,▷,◁) is a matched pair if for all x,y∈G and all a,b,∈F, x▷ab=(x▷a)((x◁a)▷c) xy◁a=(x◁(y▷a))(y◁a) While this definition turns out to be equivalent to that of a factorizable group, [Ta] did not know that when he coined this term in the 1980s. 9 Lemma 2.1.3. Let L = GF be a factorizable group. Then (F,G,▷,◁) is a matched pair where for x∈G and b∈F, the decomposition below is unique: xb=(x▷b)(x◁b). Conversely, if (F,G,▷,◁) is a matched pair, then L which has underlying set G× F and multiplication defined by (a,x)(b,y)=(a(x▷b),(x◁b)y) is a factorizable group We use the same definitions as from [JM]. Many of the results given next are proved in [JM] or follow immediately from the definitions. Proposition 2.1.4. Let L = GF = FG be a factorizable group where the actions ◁ and ▷ are defined such that xb = (x▷b)(x◁b) is unique for all x∈ G and b∈ F. Then the following properties hold: 1. x▷bc=(x▷b)((x◁b)▷c) 2. xy◁b=(x◁(y▷b))(y◁b) 3. xy▷b=x▷(y▷b) 4. x◁bc=(x◁b)◁c 5. 1▷b=b 6. 1◁b=1 7. x◁1=x 8. x▷1=1 9. (x▷b) − 1 =(x◁b)▷b − 1 10. (x◁b) − 1 =x − 1 ◁(x▷b) Both Lemma 2.1.3 and Proposition 2.1.4 are proved using direct computation from the definitions. 10 Definition 2.1.5. Let L = FG be a factorizable group. Then for x ∈ G and b ∈ F, we define the following: 1. The orbits O x ={x◁b|b∈F}⊂ G 2. the stabilizers F x ={b∈F|x◁b=x}⊂ F 3. The sets F x − 1 ,x ={b∈F|x − 1 ◁b=x} Lemma 2.1.6 ([JM], Corollary 4.3). Let L = FG. Let x ∈ G, then the following are equivalent: 1. x − 1 ∈O x 2. y − 1 ∈O x for all y∈O x 3. F x − 1 ,x ̸=∅ 4. F y − 1 ,y ̸=∅ for all y∈O x In fact, it is always true that |F x − 1 ,x |=|F y − 1 ,y | for all y∈O x . Aswewillseeinthesubsequentchapters, theheartofourproblemofcomputingsecond indicators will become a problem of understanding which x∈S n− 1 satisfy that x − 1 ∈O x . Lemma 2.1.7. Let x − 1 ∈O x . Then |F y − 1 ,y |=|F x | for every y∈O x . Proof. Since x − 1 ∈O x , there exists b∈F such that x − 1 =x◁b. Define a function φ:F x − 1 ,x →F x such that φ(c)=bc for all c∈F x − 1 ,x . This function is well defined since x◁bc=(x◁b)◁c=x − 1 ◁c=x. This map is also clearly a bijective with inverse φ − 1 (d)=b − 1 d for all d∈F x . Proposition 2.1.8 ([JM], Lemma 4.2 (4)). Let F be an abelian group and let x − 1 ∈O x . Then for all y∈O x and all b∈F y − 1 ,y , (y − 1 ▷b)b∈F x 11 The specific factorizable group of interest to us is L = S n . Letting a = (1 2 ··· n) be the standard n-cycle and considering F = C n = ⟨a⟩ and G = S n− 1 we can write L = FG as a factorizable group. Clearly C n ∩ S n− 1 = 1 since aside from the identity, permutations in S n− 1 fix n and permutations in C n do not. Furthermore, C n S n− 1 ⊂ S n and |C n ||S n− 1 | = n(n− 1)! = n! =|S n |, we have that S n = S n− 1 C n is a factorizable with F =C n and G=S n− 1 . Proposition 2.1.9. For any λ ∈S n , let a be the standard n-cycle and write λ =a r x where x = a − r λ ∈ S n− 1 and r = λ (n). This is the unique way of representing λ as a product of elements in C n S n− 1 . In Chapter 4, we will use the description of the action in Proposition 2.1.9 to better understand and reinterpret the group actions of F and G on each other. But before we can derive anything useful, we must describe the Hopf structure which is at the heart of this paper. 12 Chapter 3 Hopf Algebras 1 Hopf and Representation Results Here we construct the Hopf algebra that will be the object of study for the remainder of the paper. Throughout this work, we will take k to be an algebraically closed field of characteristic 0. When doing explicit computations, we will often take k = C. A general resource on Hopf algebras, [MO], was consulted often for background definitions and facts about semisimple Hopf algebras. The first half of this section is largely a summary of key results of [JM] since they providedtheindicatorformulaforallsemisimpleHopfalgebraswhicharebismashproducts and they also classified all indicators of our main example in the prime case. We conclude that further simplifications to their formula are possible in the case where F is abelian. Definition 3.1.1 ([Ta]). Let (F,G,◁,▷) be a matched pair of finite groups. Then the bismash product J = k G #kF has underlying set k G ⊗ k kF. Let x,y∈ G and a,b∈ F and define p x :G→k such that p x (y)=δ x,y =        1 if y =x 0 otherwise which is basis of k G dual to kG. We will typically write p x ⊗ a=p x #a for elements of J. 1. unit: 1#1 2. multiplication: (p x #a)(p y #b)=δ y,x◁a p x #ab 3. counit: ε(p x #a)=δ x,1 4. comultiplication: ∆( p x #a)= X y∈G (p xy − 1#(y▷a))⊗ (p y #a) 13 5. antipode: S(p x #a)=p (x◁a) − 1#(x▷a) − 1 We will now summarize a variety of general results in representation theory which we will be using as a starting point for computing the second indicators of J. Additionally, we will provide results on bismash products from [JM] that are of significance. Theorem 3.1.2 ([KMM]). Let J = k G #kF. Let x∈ G have orbit O x and stabilizer F x . Let V be an irreducible representation of F x , thought of as a simple kF x -module. Then ˆ V =kF ⊗ kFx V is an irreducible J-module under the action of (p y #a)· (b⊗ v)=δ y◁(ab),x (ab⊗ v) for all x ∈ G, a,b ∈ F and v ∈ V. Furthermore, every irreducible representation of J is induced from an irreducible representation of subgroups of F in this way. Explicitly, if we want to construct an irreducible representation ˆ V of the Hopf algebra J we follow the following steps. 1. Choose an element x∈G. 2. Under the matched pair action of F on G, compute the orbitO x and stabilizer F x of x. 3. F x is a subgroup of F and so has irreducible (group) representations V. 4. for each of these F x -group-representations V, there is an induced J-module ˆ V whose structure is described by Theorem 3.1.2. 5. Each distinct orbitO x will induce non-isomorphic irreducible representations of J. SinceallirrepsofJ areconstructedinthisway,thefirststepinunderstandingtheirreps ofJ (which is ultimately how we will understand the indicators) is to understand the orbits and stabilizers of the action of F on G. Lemma 3.1.3 ([KMM]). If F is abelian, the dimension of ˆ V as described in Theorem 3.1.2 is [F :F x ]. 14 In fact, in the original proof of Theorem 3.1.2, we know that dim( ˆ V)=[F :F x ]dim(V). So, if F is abelian, so is F x for all x so dim(V)=1 for all irreducible representations of F x so dim( ˆ V)=[F :F x ]. We now introduce a crucial formula, the indicator formula for bimash product Hopf Algebras. Theorem 3.1.4 ([JM]). For J =k G #kF and x∈G, the indicator of ˆ χ is given by ν (ˆ χ )= 1 |F| X y∈Ox X a∈F y − 1 ,y ˆ χ (p y #(y − 1 ▷a)a) (1.1) where F y − 1 ,y ={a∈ F|y − 1 ◁a = y}. Furthermore, if ˆ V is an irreducible representation, then ν (ˆ χ ) takes values in {1,0,− 1} Theorem 3.1.5 ([JM]). Let L = GF = FG be a factorizable group and J = k G #kF. Let x ∈ G and F x = {1}. Then there is a unique simple F x -module V ∼ = k and the induced simple J-module ˆ V has Schur indicator equal to 1 if and only if x − 1 ∈O x . Otherwise the indicator is 0. Theorem 3.1.6 ([JM]). Let x ∈ G and F x = F. Then any simple module V of kF x is already an F-module and so ˆ V =V becomes a simple J-module. Then: 1. if x = 1 then the value of the Frobenius-Schur indicator of ˆ V is the same as the indicator of V as a simple kF-module; 2. if x 2 ̸=1 then the value of the Frobenius-Schur indicator of ˆ V is 0; 3. if x 2 = 1 (for x ̸= 1) and additionally if F = C p r for p and odd prime, then the Frobenius-Schur indicator of ˆ V is 1. Now we describe a main result proved in [JM]. This is a complete classification of the indicators of the Hopf algebra J p where p is a prime. Definition 3.1.7. Let S n = S n− 1 C n = C n S n− 1 be a matched pair of finite groups. Then the bismash product J n =k S n− 1 #kC n where F =C n and G=S n− 1 . Theorem 3.1.8 ([JM]). Consider J p = k G #kC p where p > 2 is prime. There are two cases: 15 I: ˆ V has dimension 1. Here F x = F, so that x ∈ G F = {x ∈ G|x◁a = x for all a∈F}. Then either, 1. x = 1, in which case we have the trivial module ˆ V 0 with ν ( ˆ V 0 ) = 1, and p− 1 other 1-dim simple modules ˆ V, all with indicator 0; 2. x has order 2, in which case we have p simple modules with ν ( ˆ V)=1; 3. x has order >2, in which case we have p simple modules with ν ( ˆ V)=0. II: ˆ V has dimension p. Here F x ={1} and either 4. O x contains an element of order 2. Then there is one simple module with ν ( ˆ V)=1; 5. O x contains no elements of order 2. Then there is one simple module with ν ( ˆ V)=0. We now cite a slightly revised theorem of [JM]. Theorem 3.1.9 ([JM]). Consider J p =k G #kC p for p>2 prime. Let m p,1 be the number of orbits of size p (so F x ={1}) containing at least one involution. Let m p,0 be the number of orbits of size p containing no involutions. Then 1. m p,1 = i p − 1 p − 1 i p =#{σ ∈S n |σ 2 =1} 2. m p,1 +m p,0 =(p− 1) (p− 2)!− 1 p 3. There are pm p,1 irreps of dimension p with indicator +1 4. There are p(p− 1) irreps of dimension 1 and of these p+1 have indicator +1 Let us reiterate. In their paper, Jedwab and Montgomery fully describe all indicators for the Hopf algebra J p = k S p− 1 #kC p where p is an odd prime. Specifically, they describe completely all x∈G=S p− 1 for which an irrep ˆ V of J p induced from the (group) represen- tationV ofthegroupF x =(C p ) x hasnonzeroindicatorandfurthershowthatallindicators are either 0 or 1. In Chapter 7 Section 1 and Section 2, we do the same for J n for n odd and n=2 respectively. 16 Note that [JM] show that the algebraic dual (J n ) ∗ = H n = C Cn #CS n− 1 is totally orthogonal. That is, every simple H n -module admits an H n -invariant symmetric nondegen- erate bilinear form. However, we will see in the subsequent chapters that the only n for which J n is totally orthogonal is n=2. Thus, the question of describing indicators becomes much more interesting for J n . Theorem 3.1.10 ([JM]). Let H n =C Cn #CS n− 1 be the algebraic dual of J n . Then if ˆ V is an irreducible representation of H n with character ˆ χ , ν (ˆ χ )=+1. Lemma 3.1.11. Let x∈S n− 1 . If x − 1 / ∈O x , then ˆ V has indicator 0 for any representation V of F x . Proof. From Theorem 3.1.4, if F y − 1 ,y =∅, then ν (ˆ χ )=0. From Lemma 2.1.6, if x − 1 / ∈O x , then F y − 1 ,y =∅ for all y∈O x and so ν (ˆ χ )=0. Lemma 3.1.11 is a more general version of Theorem 3.1.5. Proposition3.1.12([JM]). LetL=FGasusualandletV beanirreudciblerepresentation of F x with character χ . Then for x∈ G, the induced J = k G #kF-module ˆ V has character ˆ χ which is defined by ˆ χ (p y #a)= X b∈T where b − 1 ab∈Fx δ y◁b (x)χ (b − 1 ab) where, the sum is taken over a set of representatives T for the right cosets of F/F x . Lemma 3.1.13. If F is abelian, then the formula for ˆ χ From Proposition 3.1.12 reduces to ˆ χ (p y #a)=        χ (a) if y∈O x and a∈F x 0 otherwise Proof. Fix the element p y #a. Consider the following cases. 1. First, if y / ∈ O x , then there does not exist any b ∈ F for which y◁b = x. Namely, δ y◁b (x)=0 for all b∈F and so ˆ χ (p y #a)=0. 2. Second,ify∈O x buta / ∈F x ,thenthereisnob∈F forwhichbab − 1 =bb − 1 a=a / ∈F x (F is abelian), so there is no sum and thus ˆ χ (p y #a)=0. 17 3. Now, if y ∈O x and a∈ F x , then by Proposition 3.1.12, we claim that there is only one coset of F/F x for which y◁b=x. First, since y ∈ O x = {x◁ b|b ∈ F} there exists b ∈ F so x◁ b = y. Namely, x = y◁b − 1 and so there is at least one coset (b − 1 F x ) of F/F x whose representative satisfies that δ y◁b − 1(x)=1. Now, assume c ∈ F is another element such that y◁ c = x (and so equivalently x◁c − 1 =y). Then y =x◁c − 1 =x◁b =⇒ x=x◁bc. Thus, bc∈F x so c∈b − 1 F x (keep in mind F is abelian). Thus, there is exactly one coset for which δ y◁b − 1(x)̸=0. Finally, we obtain the result ˆ χ (p y #a)=        χ (a) if y∈O x and a∈F x 0 otherwise Theorem 3.1.14. Let C n = ⟨a⟩. Let k = C and ζ n ∈ k be a primitive n th -root of unity. Let V i be a simple kC n -module equivalent to the irreducible (group) representation ρ i of C n defined by ρ i :C n →C × a7→ζ i n . Then, for any 0≤ i<n, ν (V i )=        +1 if n|2i 0 otherwise Finally, this gives us that for n even, all the representations have indicator 0 except for exactly the trivial representation and the representation defined by ρ i (a) = (ζ n ) n 2 both of which have indicator +1. And for n odd, all representations have indicator 0 except the trivial one which has indicator +1. 18 Proof. Let χ i be the character of V i . First, note that V i ∼ = k and so writing ρ i (a) = ζ i n makes sense and so Tr(ρ i (a))=ζ i n . Since C n =⟨a⟩, we can write ν (V i )= 1 |C n | X b∈Cn χ i (b 2 )= 1 |C n | n− 1 X t=0 χ i ((a t ) 2 ) = 1 n n− 1 X t=0 Tr(ρ i (a 2t ))= 1 n n− 1 X t=0 ρ i (a) 2t = 1 n n− 1 X t=0 (ζ i n ) 2t = 1 n n− 1 X t=0 ζ 2it n =        1 if ζ 2i n =1 1 n · ζ 2ni n − 1 ζ 2i n − 1 if ζ 2i n ̸=1 =        1 if ζ 2i n =1 0 if ζ 2i n ̸=1 Recall that for any w ̸= 1, w n − 1 w− 1 = n− 1 X t=0 w t . Since ζ n is a primitive n th -root of unity, ζ 2i n =1 if and only if n|2i. Note that since i < n, 2i < 2n and so n|2i if and only if n = 2i (so i = n 2 ) or i = 0. Namely, the trivial representation always has indicator +1, and there is a non-trivial representation with nonzero indicator if and only if n is even. 19 Chapter 4 Understanding the Action 1 Reinterpreting the Action Consider x∈S n such that x(n)=n. Then, letting C n =⟨a⟩ where a=(1 2 ··· n) is the standard n-cycle, we can multiply powers of a with x without issue. Now, we note that S n ∼ = Bij(Z/nZ) which is the group of bijections of Z/nZ where permutation multiplication is now viewed as function composition. By viewing a as a shifting map which sends i to i+1, we can consider how the action of C n on S n− 1 affects each permutation x pointwise. Using this notation, we will often describe addition and multiplication of the outputs of x and its shifts. This makes sense using the ring structure of Z/nZ. Unless otherwise stated, all arithmetic in this chapter will be done modulo n. 2 Reinterpreting Orbits and Stabilizers Proposition 4.2.1. Let L =FG =C n S n− 1 where F =C n =⟨a⟩ and G =S n− 1 . consider the actions◁ and▷ of F and G on each other as from Section 1 and let x∈S n− 1 . Then, 1. Let a r ∈C n . Then x◁a r =a − x(r) xa r and x▷a r =a x(r) ; 2. F x ={a t |a − x(t) xa t =x}={a t |x(u+t)− x(t)=x(u) for u=1,...,n}; 3. F x − 1 ,x ={a s |x − 1 (u+s)− x − 1 (s)=x(u) for i=1,...,n} 4. O x ={a − x(l) xa l |a l ∈C n }; Proof. Note that all arithmetic is done modulo n. That is, when we write x(u+t)− x(t)= x(u) we really mean (x(u+t mod n)− x(t)) mod n=x(u). 1. Proposition2.1.9tellsusthatS n isfactorizablewithsubgroupsF =C n andG=S n− 1 and Lemma 2.1.3 tells us how we can obtain a matched pair from F and G. 20 2. Recall Definition 2.1.5 Item 2. From (1) F x ={a t ∈C n |x◁a t =x} ={a t ∈C n |a − x(t) xa t =x} ={a t ∈C n |(a − x(t) xa t )(u)=x(u) for all u=1,...,n} ={a t ∈C n |x(u+t)− x(t)=x(u) for allu=1,...,n} 3. Recall Definition 2.1.5 Item 3. Then, by similar reasoning as (2), F x − 1 ,x ={a s ∈C n |x − 1 ◁a s =x} ={a s ∈C n |a − x − 1 (s) x − 1 a s =x} ={a s ∈C n |(a − x − 1 (s) x − 1 a s )(u)=x(u) for all u=1,...,n} ={a s ∈C n |x − 1 (u+s)− x − 1 (s)=x(u) for allu=1,...,n} 4. Recall Definition 2.1.5 Item 1. Again from (1) O x ={x◁a l |a l ∈C n } ={a − x(l) xa l | for l =1,...,n} Example 4.2.2. Let n = 6 and x = (1 2)(3 4 5). Then, using permutation multiplication we get that x◁ a = a − x(1) xa = a − 2 xa = a 4 xa. Since a = (1 2 3 4 5 6), direct computation by multiplying cycles gives that x◁a=(1 5 4). However, instead of multiplying permutations, this result can also be seen by understanding x◁a pointwise. That is, using ideas from the proof of Proposition 4.2.1 Item 1, (x◁a)(i)=(a 4 xa)(i)=x(i+1)+4. This allows us to write out the product as a bijection (which aligns with direct computation). 17→x(2)+4=5 27→x(3)+4=2 37→x(4)+4=3 47→x(5)+4=1 57→x(6)+4=4 67→x(1)+4=6 21 Proposition 4.2.3. Let x∈S n− 1 . Let F x =⟨a t ⟩. Then 1. a − x(l) xa l =a − x(s) xa s if and only if x(s− l)=x(s)− x(l) and s=l mod t 2. O x ={a − x(l) xa l |l =1,...,t} and so if y =a − x(l) xa l ∈O x , then y(u)=x(u+l)− x(l) for all u. Specifically, F y =F x and y(t)=x(t). 3. If x − 1 = a − x(s) xa s ∈O x . Then x(s)+s is a fixed point of x and x(s)+s = ut for some u. 4. If x − 1 =a − x(s) xa s ∈O x , and we let y∈O x with y =a − x(l) xa l . Then F y − 1 ,y ={a mt+l− s− x(l) |m=0,1,..., n t − 1} Proof. First, note that Proposition 4.2.1 Item 2 tells us that if a t ∈ F x , then x(t+u) = x(t)+x(u) for all u. 1. =⇒ Let a − x(l) xa l =a − x(s) xa s for some l and s. Then x=a x(l)− x(s) xa s− l . Now, x(t)=(a x(l)− x(s) xa s− l )(t)=x(t+s− l)+x(l)− x(s)=x(t)+x(s− l)+x(l)− x(s) and so − x(s− l)=x(l)− x(s). Thus, x=a x(l)− x(s) xa s− l =a − x(s− l) xa s− l so a s− l ∈F x so there exists a p such that s− l =pt so s=l mod t. ⇐= Assume there exists s,l so x(s− l)=x(s)− x(l) and s=l mod t. Let s− l =pt for some p. Then, since a − pt = (a t ) − p ∈ F x , Proposition 4.2.1 Item 2 gives that a − x(− pt) xa − pt =x. Using the substitution l =s− pt, routine computation gives that a − x(l) xa l =a − x(s) xa s as desired. 2. From Proposition 4.2.1 we know that O x = {a − x(l) xa l |l = 1,...,n}. Let y ∈ O x . Then y =a − x(l) xa l for some l. Thus, y(t)=(a − x(l) xa l )(t)=x(t+l)− x(l)=x(t)+x(l)− x(l)=x(t). 22 Furthermore, a − y(t) ya t =a − x(t) ya t =a − x(t) (a − x(l) xa l )a t =a − x(l) a − x(t) xa t a l =a − x(l) xa l =y. Therefore, a t ∈F y and so F x ⊂ F y . However,O x =O y bydefinitionofgroupactionsthechoiceofrepresentativedoesnot change the orbit and so this shows that stabilizers are contained within each other. Thus, F x =F y for all y∈O x . Finally, let O x ={x,y 1 ,y 2 ,...,y n− 1 } where y l = a − x(l) xa l . Now, since the orbit stabilizer theorem tells us that |O x | = |F| |Fx| = n n t = t If n = t, then there is nothing to show. However, if t < n, then consider p=ut+q for some q <t. Then inductively, x(ut+q)=x(ut)+x(q) and so x(ut+q− q) = x(ut) = x(ut+q)− x(q). Clearly ut+q = q mod t and so (1) gives us that y p =y q . Therefore, O x ={x,y 1 ,y 2 ,...,y n− 1 }={x,y 1 ,...,y t− 1 ,x,y 1 ,...y t− 1 ,...}={x,y 1 ,...,y t− 1 }. That is,O x ={a − x(l) xa l |l =1,...,t} 3. Let x − 1 =a − x(s) xa s for some s≤ t. Then we have that x − 1 (u)=x(u+s)− x(s) for all u. Namely, x(x(l)+s)=x − 1 (x(l))+x(s)=l+x(s) and so Then s=x − 1 (x(s))=x(x(s)+s)− x(s) =⇒ x(x(s)+s)=x(s)+s so x(s)+s is a fixed point of x. Furthermore, a − x(x(s)+s) xa x(s)+s =a − (x(s)− s) xa x(s)+s =a − s (a − x(s) xa s )a x(s) =a − s x − 1 a x(s) =(a − x(s) xa s ) − 1 =(x − 1 ) − 1 =x so a x(s)+s ∈F x so x(s)+s=ut for some 1≤ u≤ n t . 23 4. Let x − 1 =a − x(s) xa s and y∈O x with y =a − x(l) xa l . Then y − 1 =(a − x(l) x l ) − 1 =a − l x − 1 a x(l) =a − l a − x(s) xa s a x(l) =a − l− x(s) xa x(l)+s Note that (3) gives that y − 1 =a − x(x(l)+s) xa x(l)+s . Next, let a r ∈F y − 1 ,y . Note that since we are assuming that x − 1 ∈O x , we know that F y − 1 ,y is nonempty and has the same size as F x from Lemma 2.1.6. Then, y =a − x(l) xa l =y − 1 ◁a r =a − y − 1 (r) y − 1 a r =a − y − 1 (r) (a − x(x(l)+s) xa x(l)+s )a r =a − x(r+x(l)+s)+x(x(l)+s) a − x(x(l)+s) xa x(l)+s a r =a − x(x(l)+s+r) xa x(l)+s+r Thus, we have that a − x(l) xa l =a − x(x(l)+s+r) xa x(l)+s+r and so from (1), there exists a p such that x(l) + s + r− l = pt. Namely, a r = a pt+l− x(l)− s . Conversely, fix 1 ≤ m≤ n t and let r =mt+l− x(l)− s. Then y − 1 (r)=y − 1 (mt+l− x(l)− s)=x(mt+l)− l− x(s)=mx(t)+x(l)− l− x(s) and y − 1 ◁a r =a − y − 1 (r) y − 1 a r =a − y − 1 (r) (a − l− x(s) xa x(l)+s )a r =a − mx(t)− x(l)+l+x(s) a − l− x(s) xa x(l)+s a r =a − x(l)− x(mt) xa x(l)+s+r =a − x(l) a − x(mt) xa mt+l =a − x(l) xa l =y and so a mt+l− x(l)− s ∈F y − 1 ,y for all m. Therefore, F y − 1 ,y ={a mt+l− x(l)− s |m=1,2,..., n t }. 24 Sincewearetakingallintegersmodulon,wecouldequivalentlyconsideru=0,..., n t − 1, which will be convenient to us later. 3 Uncovering New Permutations Properties Now, using our new notation, we will explore exactly how much we can obtain from this shifting. It turns out the answer is quite a lot! First, we dive into understanding what info we can obtain from the stabilizer of F x ⊂ F =C n . Because the stabilizer is a subgroup and C n is cyclic, we can write F x =⟨a t ⟩ for some value t dividing n. Throughout, we will refer to t often and it will always be reserved for the (smallest) power of a which generates the stabilizer of a given x. Lemma 4.3.1. Let x∈S n− 1 and F x =⟨a t ⟩ where a=(1 2 ··· n). Then, 1. x(ut+q)=ux(t)+x(q) for all u∈Z/nZ and q <t. 2. x(t)=jt where 1≤ j < n t is coprime to n t 3. F x − 1 =F x 4. If x(t)=jt then x − 1 (t)=j − 1 t where j − 1 is the multiplicative inverse of j modulo n t . 5. if x − 1 ∈O x and x(t)=jt, then j 2 =1 mod n t . Proof. 1. Thisisinductiononthestatmentx(u+t)=x(u)+x(t)foralltfromProposition4.2.1 Item 2. 2. Here we are assuming that F x =⟨a t ⟩⊂ F =⟨a⟩. Since|F|=n, andF x is a subgroup, Lagrange tells us that there exists an m such that n=mt. Therefore by (1), mt=n=x(n)=x(mt)=mx(t). Namely, using rules of modular arithmetic, mx(t)=mt mod n =⇒ x(t)=t mod n m =t . Thus, x(t) is some multiple of t. 25 Now, for contradiction, assume that x(t)=jt where j is not coprime to n t . Then, let u= n tgcd(j, n t ) < n t and so ut<n but x(ut)=ux(t)=ujt= njt tgcd(j, n t ) =n j gcd(j, n t ) =n mod n which is not possible since x∈ S n− 1 and so x(n) = n. Namely, x(t) = jt where j is coprime to n t . 3. SinceProposition4.2.1Item1givesusthatx(u+t)=x(u)+x(t)forallu,inductively we get that x(mt+u)=x((m− 1)t+u+t)=x((m− 1)t+u)+x(t)=··· =x(u)+mx(t). 4. First, since x◁a t =x, then inverting both sides and using the properties of Proposi- tion 2.1.9 and Proposition 4.2.1 Item 1, we obtain (x◁a t ) − 1 =x − 1 ◁(x▷a t )=x − 1 ◁a x(t) =x − 1 . Namely, a x(t) ∈ F x − 1. However, x(t) = jt where j is coprime to n t by (2). Thus, a x(t) =(a t ) j is a generator of F x and so F x ⊂ F x − 1. Now, the reverse. Let F x − 1 =⟨a s ⟩, then x − 1 ◁a s =x − 1 so again using the properties of Proposition 2.1.9 and Proposition 4.2.1 Item 1, (x − 1 ◁a s ) − 1 =x◁(x − 1 ▷a s )=x◁a x − 1 (s) =x so a x − 1 (s) ∈ F x . But (2) tells us that x − 1 (s) = ls where l is coprime to n s and so again, a x − 1 (s) is a power of the generator which is coprime to n s =|F x − 1| and so it is again a generator. Thus, F x − 1 ⊂ F x . 5. Note that from (2), we know that x(t) = jt where j is coprime to n t . Thus, j has a unique multiplicative inverse modulo n t which we will call j − 1 . Let x − 1 (t)=lt where l is coprime to n t . Then, from (1) t=x − 1 (x(t))=x − 1 (jt)=jx − 1 (t)=jlt =⇒ 1=jl mod n t . Thus, because inverses (if they exist) are unique, we have that l =j − 1 mod n t . 26 6. If there is an s so x − 1 = a − x(s) xa s ∈ O x , then x − 1 (u) = x(u + s)− x(s) for all u=1,...,n− 1. Namely, x − 1 (t)=x(t+s)− x(s)=x(s)+x(t)− x(s)=x(t). That is, t=x 2 (t)=x(jt)=jx(t)=j 2 t and so j 2 t=t mod n =⇒ j 2 =1 mod n t . Remark 4.3.2. The converse of Lemma 4.3.1 Item 5 is false. That is, knowing x satisfies that x(t)=jt where j 2 =1 mod n t does not force x − 1 ∈O x . Corollary 4.3.3. Let x∈S n− 1 . Then a t ∈F x if and only if x(mt+u)=mx(t)+x(u) for all 1≤ m≤ n t and 0≤ u<t. Proof. =⇒ This follows directly from Lemma 4.3.1 Item 1. ⇐= Assume x(mt+u) = mx(t)+x(u) for all 1≤ m≤ n t and 0≤ u < t. Let 1≤ q≤ n and write q =mt+u for some 0≤ u<t. Then, x(q+t)− x(t)=x(mt+u+t)− x(t)=x((m+1)t+u)− x(t) =(m+1)x(t)+x(u)− x(t)=mx(t)+x(u) =x(mt+u)=x(q) So a t ∈F x by Proposition 4.2.3. Corollary 4.3.4. Let x∈S n− 1 . Then F x =F =C n =⟨a⟩ if and only if for all u=1,...,n, x(u)=ux(1). Let us pause here to emphasize the volume of information we obtain from knowing the size of the stabilizer of x. Lemma 4.3.1 tells us that if F x =⟨a t ⟩, then x is in some sense linear in t. That is, x(q) where q > t is determined by where x sends q mod t and where x sends t. This puts a heavy restriction on what x can look like. All values x(q) for q > t are determined from the values x(t) and x(u) for u < t. Thus, if t is small relative to n, then 27 the number of permutations in S n− 1 which have stabilizer⟨a t ⟩ is small relative to the total (n− 1)! which we formalize in Chapter 6. Using our new understanding of permutations x satisfying F x =⟨a t ⟩, we can now better understand the relationship between x and y where y∈O x . Lemma 4.3.5. Let x ∈ S n− 1 , F x = ⟨a t ⟩, and x − 1 ∈ O x . Write O x = {x,y 1 ,y 2 ,...,y t− 1 } where y l =a − x(l) xa l . Then 1. If x − 1 = y s for 1 ≤ s ≤ t, then y − 1 l = y x(l)+s mod t for all l = 1,...,t. (Here we let y t =x). 2. for some 1 ≤ l ≤ t, l = x(l)+s mod t if and only if y l has order 2 (Here we let y t =x). Proof. 1. If x − 1 =y s =a − x(s) xa s for some s≤ t, then y − 1 l =(a − x(l) xa l ) − 1 =a − l x − 1 a x(l) =a − l a − x(s) xa s a x(l) =a − l− x(s) xa s+x(l) Now note that x − 1 =a − x(s) xa s implies that for all j, x − 1 (j)=x(j+s)− x(s) and so − x(x(l)+s)=− (x − 1 (x(l))+x(s))=− (l+x(s))=− l− x(s) and so we obtain that y − 1 l =a − x(x(l)+s) xa x(l)+s =y x(l)+s mod t . 2. This is immediate from (1) since y l has order two if and only if y − 1 l = y x(l)+s = y l which happens if and only if x(l)+s=l mod t. 28 Chapter 5 Indicators of J n 1 A Revised Indicator Formula Proposition 5.1.1. Let x∈S n− 1 , F x =⟨a t ⟩, x − 1 =a − x(s) xa s ∈O x for some s, x(t)+t= u 1 t, and x(s)+s=u 2 t. Fix an integer i. If iu 1 =0 mod n t then 1. if n t is odd, iu 2 =0 mod n t 2. if n t is even, then iu 2 = n 2t or 0 mod n t . Proof. First, note that since x − 1 (i)=x(i+s)− x(s), letting i=x(s) we get that x(x(s)+s)=x − 1 (x(s))+x(s)=s+x(s). Now, a − x(x(s)+s) xa x(s)+s =a − (x(s)+s) xa x(s)+s =a − s a − x(s) xa s a x(s) =a − s x − 1 a x(s) =(a − x(s) xa s ) − 1 =(x − 1 ) − 1 =x. Therefore, a x(s)+s ∈F x =⟨a t ⟩ and so x(s)+s=u 2 t is a multiple of t. Thus, u 2 x(t)=x(u 2 t)=x(x(s)+s)=x(s)+s=u 2 t and so 2u 2 t=u 2 t+u 2 t=u 2 x(t)+u 2 t=u 1 u 2 t. This gives us the relation that 2u 2 t=u 1 u 2 t mod n =⇒ 2u 2 =u 1 u 2 mod n t . Therefore, 2iu 2 =iu 1 u 2 =0 mod n t . 1. Assume n t is odd. Then 2 is invertible mod n t so iu 2 =0 mod n t . 29 2. Assume n t is even. Then iu 2 =0 mod n 2t . We are at last ready to give the key theorem so far. Theorem 5.1.2. Consider J n =C S n− 1 #CC n . Let C n =⟨a⟩. Fix x∈S n− 1 such that F x = ⟨a t ⟩ and x − 1 =x◁a s ∈O x . Let ρ x,i :F x →C × be an irreducible (group) representation of F x defined by ρ x,i (a t )=ζ i n t where ζ n t is a primitive n t -root of unity. Let χ x,i be the character of ρ x,i Then if ˆ χ x,i is the character of J n induced from χ x,i (as described in Theorem 3.1.2) we obtain: 1. if n t is odd, then ν (ˆ χ x,i )=        1 if (ζ n t ) iu 1 =1 0 if (ζ n t ) iu 1 ̸=1 (1.1) 2. if n t is even, then ν (ˆ χ x,i )=        (ζ n t ) − iu 2 =1 or − 1 if (ζ n t ) iu 1 =1 0 if (ζ n t ) iu 1 ̸=1 (1.2) where x(t)+t=u 1 t and x(s)+s=u 2 t. Proof. Let ρ x,i : F x → C × be an irreducible (group) representation of F x defined by ρ x,i (a t ) = (ζ n t ) i where ζ n t is a primitive n t -root of unity. Let χ x,i be the character of V x,i which is the F x -module equivalent to ρ x,i . Then if ˆ χ x,i is the character of J n induced from χ x,i (as described in Theorem 3.1.2) First,recallEquation(1.1). NotethatProposition2.1.8tellsusimmediatelythatassum- ing x − 1 ∈O x gives that ˆ χ x,i (p y #b)=χ x,i (b) and so we get ν (ˆ χ x,i )= 1 |F| X y∈Ox X b∈F y − 1 ,y χ x,i ((y − 1 ▷b)b) where χ x,i is the character of V x,i which is the F x -module which induces ˆ χ x,i . Since x − 1 ∈O x , let x − 1 =a − x(s) xa s for some s≤ t. 30 Let y l ∈O x with y l =a − x(l) xa l . Then Proposition 4.2.3 Item 4 gives that we can write b∈F y − 1 ,y as b=a mt+l− s− x(l) for some 1≤ m≤ n t . Note that we have that y − 1 l =(a − x(l) xa l ) − 1 =a − l x − 1 a x(l) =a − l (a − x(s) xa s )a x(l) and so y − 1 l (mt+l− s− x(l))=x(mt+l)− x(s)− l, Thus, (y − 1 l ▷b)b=a y − 1 l (mt+l− s− x(l)) a mt+l− s− x(l) =a x(mt+l)− x(s)− l a mt+l− s− x(l) =a x(mt)+mt− x(s)− s =a m(x(t)+t)− (x(s)+s) Note that (y − 1 ◁b) is very much dependent on both y and b. However, (y − 1 ◁b)b is dependent on only the choice of the s for which x − 1 =a − x(s) xa s . Let x(t) + t = u 1 t and x(s) + s = u 2 t. Note that both these sums are necessarily multiplies of t since x(t) is a multiple of t from Lemma 4.3.1 Item 2 and x(s) + s is a multiple of t from Proposition 4.2.3 Item 4. Then, ν (ˆ χ x,i )= 1 |F| X y∈Ox X b∈F y − 1 ,y χ x,i ((y − 1 ▷b)b) = 1 n t X l=1 n t − 1 X m=0 χ x,i ((y − 1 l ▷a mt+l− s− x(l) )a mt+l− s− x(l) ) = 1 n t X l=1 n t − 1 X m=0 χ x,i (a m(x(t)+t)− (x(s)+s) ) = t n n t − 1 X m=0 χ x,i ((a t ) mu 1 − u 2 ) = t n n t − 1 X m=0 (ζ n t ) i(mu 1 − u 2 ) = t n n t − 1 X m=0 (ζ n t ) imu 1 (ζ n t ) − iu 2 31 = t n (ζ n t ) − iu 2        n t if (ζ n t ) iu 1 =1 (ζ n t ) iu 1 n t− 1 (ζ n t ) iu 1− 1 if (ζ n t ) iu 1 ̸=1 =        (ζ n t ) − iu 2 if (ζ n t ) iu 1 =1 t n (ζ n t ) − iu 2 (ζ n t ) iu 1 n t− 1 (ζ n t ) iu 1− 1 if (ζ n t ) iu 1 ̸=1 =        (ζ n t ) − iu 2 if (ζ n t ) iu 1 =1 0 if (ζ n t ) iu 1 ̸=1 with the last line reducing since ζ n t is a n t th -root of unity, so (ζ n t ) iu 1 n t =((ζ n t ) n t ) iu 1 =1 so (ζ n t ) iu 1 n t − 1=0. 1. Finally, Proposition 5.1.1 gives us that iu 2 is a multiple of n t if n t is odd, and so (ζ n t ) iu 2 =1 in this case. 2. In the other case, iu 2 is a multiple of n 2t if n t is even. Namely, there exists an m so iu 2 =m n 2t . Therefore, (ζ n t ) iu 2 =(− 1) m which could be− 1 if m is odd. Note that Theorem 5.1.2 agrees completely with the results of [JM]. Namely, Theo- rem 3.1.5 tells us that if F x ={1} then ν (ˆ χ )=1 if and only if x − 1 ∈O x and is 0 otherwise. Theorem 5.1.2 says that if F x ={1} then t = n so n t = 1 which is odd. There is then one (group) irrep of F x , the trivial (group) irrep and so the indicator will be 1 if x − 1 ∈O x and 0 otherwise. Theorem 3.1.6 Item 2 says that if F x = F and x 2 ̸= 1 then all indicators are 0. Theo- rem 5.1.2 says that if F x = F then t = 1 and so n t = n. Since|O x | = t = 1, if x 2 ̸= 1 then x − 1 ̸∈O x ={x} and so all indicators are 0. Theorem3.1.6Item1tellsusthatifx=1,thentheindicatorof[JM]andtheFrobenius- Schur group indicator agree. Theorem 5.1.2 states that if x = 1 then x − 1 = x◁a t and so s = t. Namely, (ζ n ) iu 1 = (ζ n ) 2i and implies (ζ n ) − iu 2 = (ζ n ) − 2i . Thus, whether n is even or odd, if ζ 2i n =1 then ζ − 2i n =1 and so the indicator ν (ˆ χ x,i )=1 if and only if n divides 2i and 0 otherwise. This of course agrees with the indicator ν (χ x,i ) by Theorem 3.1.14 and so the indicators are the same. 32 Finally, we will see in the later chapters that Theorem 5.1.2 agrees with the work of [JM] when n=p r for p and odd prime and r >0. Corollary 5.1.3. If n is odd, then all indicators of J n are nonnegative. Animportantquestioniswhethertheindicatorisevernegative. Recallthatfor pprime, J p has nonnegative indicator. However, the answer is yes, there exists even n for which J n has negative indicator. Specifically, [S2] gives an example for n = 12. Here is an example for n=16. Example 5.1.4. Let n=16, one can show the element x=(1 5 9 13)(3 7 11 15) has stabilizer F x =⟨a 2 ⟩ and so t=2. Now, let ζ n t =e 2πi 8 ∈C × be a primitive 8 th -root of unity. Then, n t =8 is certainly even and by theorem, Theorem 5.1.2, ν (ˆ χ )=        ζ − 3i 8 if ζ 2i 8 =1 0 if ζ 2i 8 ̸=1 Finally, consider the irreducible (group) representation of F x , ρ 4 defined by ρ 4 (a 2 )=ζ 4 8 . That is, we can let i=4. Then ζ 2i 8 =ζ 8 8 =1 and so we have that ζ 4 8 =e 4 2πi 8 =e πi =− 1 and so ν (ˆ χ )=ζ − 3i 8 =ζ − 12 8 =(ζ 4 8 ) − 3 =(− 1) − 3 =− 1. In fact, we will see that there exists irreps of J n which have negative indicator for any n≥ 12 with 4|n. Before we conclude this section, we provide two important results that we will use in later chapters. Proposition 5.1.5. Let x ∈ S n− 1 , F x = ⟨a t ⟩, x − 1 = a − x(s) xa s ∈ O x , and x(t) = jt for some j coprime to n t . Let ˆ χ be the character induced from an irreducible representation of F x . Then, 33 1. Permutations in the same orbit have the same indicators. That is, if y ∈ O x then for a given representation, the induced characters ˆ χ x and ˆ χ y have the same indicator. (This is true even if x − 1 ̸∈O x ). 2. If O x contains an element of order 2, then ν (ˆ χ )≥ 0 3. If t is odd, then ν (ˆ χ )≥ 0. 4. If n=2m for m odd then ν (ˆ χ )≥ 0. Proof. 1. This is a result from [KMM]. If x and y are in the same orbit, then the irreps of J induced from x and y are isomorphic. 2. If y∈O x has order 2, then s = t and so if (ζ n t ) i y(t)+t t = 1 then clearly (ζ n t ) − i y(s)+s t = ((ζ n t ) i y(t)+t t ) − 1 =1 − 1 =1 as well. By (1), the indicator of the representation induced by y is the same as that of x and so the indicators are all nonnegative. 3. If t is odd, then |O x | = |F| |Fx| = n n t = t and x − 1 ∈O x , then every y ∈O x also has its inverseinO x andsoO x containsaninvolution. By(2),allindicatorsarenonnegative. 4. If n = 2m where m is odd, then either t is odd, in which case (3) tells us ν (ˆ χ )≥ 0, or n t is odd, in which case Theorem 5.1.2 tells us ν (ˆ χ )≥ 0. Proposition 5.1.6. The only n for which J n is totally orthogonal is n=2. Proof. 1. J n can only be defined for n≥ 2 since its structure is derived from S n− 1 and S m is not defined for m < 1. Letting n = 2, gives that S 1 = {(1)} so the only possible permutation is the identity x = (1) and so F x = F. In this case, we have that t = 1. Now, we just plug into Theorem 5.1.2. u 1 = x(1)+1 1 = 2 so if ζ 2 is a primitive square root of unity, then ζ 2i 2 =1 for all i=0,1, we have that ν (ˆ χ (1),i )=+1. 2. Now, let n > 2. Again, consider the identity permutation (1) ∈ S n− 1 . Then of course, F x = F again, and u 1 = 2, but now ζ 2i n = 1 if and only if 2i = 0 mod n. If 34 n is odd, i = 0 is the only solution, and if n is even then i = 0, n 2 are both solutions. Since n > 2, there exists 0 ≤ i ≤ n− 1 such that 2i ̸= 0 and so for these choices, ν (ˆ χ (1),i ) = 0. Therefore, not all the indicators are +1 and so J n is not totally orthogonal for any n>2. 35 Chapter 6 Prerequisite Counting Recursions Of course, the goal for the project is to try and count explicitly the number of irreps of J n which have indicator +1,− 1, and 0 for a given dimension. This result is very difficult to obtaindirectlysincethereare(n− 1)!permutationsandwemustcheckthestabilizerofeach permutation in order to determine how many associated representations are induced from x (see Theorem 3.1.2). However, using our new understanding of the group actions which describe the structure of J n and our new reduced formulas from the preceding chapters, we are able to remove a lot of extraneous computation. Throughoutthischapter,allarithmeticwillbedonemodulonunlessotherwisespecified. For example, when we write x(2t)=2x(t) we mean x(2t mod n)=2x(t) mod n. For us, an involution is any permutation x∈ S n− 1 which is its own inverse (including the identity). Additionally, we will frequently invoke a result from elementary number theory: The equation ax = b mod n has either no solutions, or gcd(a,n) solutions. Specifically, the equation ax=0 mod n always has gcd(a,n) solutions. 1 Structure of Permutations with Given Stabilizer As we have seen from the previous chapters, if t is odd, then all t-dimensional irreps of J n have nonnegative indicators (Proposition 5.1.5). Furthermore, the only way an (odd) t-dimensional irrep of J n can have an indicator of +1 is if the permutation x which induces that irrep shares its orbit with an involution. Therefore, describing the odd dimensional irreps as having indicator either +1 or 0 is equivalent to understanding and counting the orbits under the matched pair action which contain involutions. 36 Proposition 6.1.1. Let M n/t be the set of x∈ S n− 1 such that F x =⟨a t ⟩. Let φ(n) be the Euler Totient Function which is the number of integers coprime to n. Then |M n/1 |=φ(n) (1.1) |M n/t |=φ n t n t t− 1 (t− 1)!− X s|t s̸=t |M n/s | (1.2) Proof. Let φ(n)={1≤ j≤ n|j coprime to n} be the Euler Totient function. The proof is simple counting. Corollary 4.3.4 tells us that F x = F if and only if x(u)=ux(1), Lemma 4.3.1 says that x(1)=j for some j coprime to n, and Corollary 4.3.3 statesthatxisdeterminedbywhereitsends1. Thus,thenumberofpermutationsx∈S n− 1 withfullstabilizerisgivenbythenumberofchoicesof x(1), whichisthenumberofintegers coprime to n. Therefore,|M n/1 |=φ(n). Now, let t>1 be a divisor of n. Again, Corollary 4.3.3 tell us that if we want to count all permutations x ∈ S n− 1 with stabilizer F x = ⟨a t ⟩, then to fully determine x, we need only understand where it sends 1,2,3,...,t. Since x(t) = jt where j is coprime to n t by Lemma 4.3.1, we know there are φ( n t ) choices for where x can send t. Now,wherecanxsend1? Wellitcertainlycannotsendittoanymultipleof tsincethen x would no longer be a bijection. Thus, it can send 1 to any number u where 1≤ u≤ n− 1 and u is not a multiple of t. There are n− n t =(t− 1) n t such options. Recall that once we know x(1), we also know x(1 + t), x(1 + 2t),...,x(1 + ( n t − 1)t). Namely, once we choose x(1) we have fixed n t outputs. Similarly, x can send 2 to any available value. Since all multiples of t and now all values of the form qt+x(1) for 1 ≤ q ≤ n t are taken. So there are n− 2 n t = (t− 2) n t possible options for x(2). Inductively we obtain: φ n t | {z } choices for x(t) · (t− 1) n t | {z } choices for x(1) · (t− 2) n t | {z } choices for x(2) ··· n t |{z} choices for x(t− 1) . Now, define x(qt+u)=qx(t)+x(u) and because this is a necessary condition we have that a t ∈F x by Corollary 4.3.3. However, we will have over counted here. Because this now counts all x ∈ S n− 1 for which a t ∈F x , and it is not necessarily true that F x =⟨a t ⟩ for each of these permutations. 37 Namely, we must now remove all x for which F x =⟨a s ⟩ where s divides t. That is, we have over counted by exactly|M n/s | for each s which is a divisor of t. Finally, this gives the desired result, |M n/t |=φ n t (t− 1) n t (t− 2) n t ··· n t − X s|t s̸=t M n/s =φ n t n t t− 1 (t− 1)!− X s|t s̸=t |M n/s | Remark 6.1.2. For example, when n=12 |M 12/1 |=4 |M 12/2 |=2(6)1!− 4=8 |M 12/3 |=2(4) 2 2!− 4=60 |M 12/6 |=1(2) 5 5!− 4− 8− 60=3,768 |M 12/12 |=1(1) 9 9!− 4− 8− 60− 3768=359,040 In fact, Proposition 6.1.1 gives a recursive formula for explicitly constructing all permu- taitons x which have stabilizer F x =⟨a t ⟩ for a given t. Using this formula, we will show in Chapter 7, Section 1 that for a fixed t, the ratio |M n/t |/(n− 1)!→0asn→∞.Thisshowsthatitisrareforapermutationtohavestabilizer ⟨a t ⟩ for large n. Corollary 6.1.3. The number of orbits of size t is |M n/t | t . Now, let us describe explicitly how to write x in terms of a permutation in a smaller symmetric group. Lemma 6.1.4. Let x ∈ S n− 1 with F x = ⟨a t ⟩. Then there exists an integer j which is coprime to n t (and so therefore invertible modulo n t ), a permutation σ x ∈S t− 1 and integers 1≤ u i ≤ n t such that x(i)=u i t+σ x (i) mod n for all 1≤ i≤ t− 1 x(qt)=qjt mod n for all 1≤ q≤ n t 38 x(qt+i)=qx(t)+x(i) mod n Furthermore, x − 1 (i)=− ju σ − 1 x (i) t+σ − 1 x (i) mod n for all 1≤ i≤ t− 1 x − 1 (t)=j − 1 t mod n x − 1 (qt+i)=qx − 1 (t)+x − 1 (i) mod n We call σ x a remainder permutation for x. Proof. Let x ∈ S n− 1 with stabilizer F x = ⟨a t ⟩. We have already established in Corol- lary 4.3.3 that x is determined by where it sends 1,...,t. We now claim that x defines a permutation modulo t. From, Lemma 4.3.1, there exists a j coprime to n t (and so invertible modulo n t ) with x(t) = jt. Now, if i,i ′ < t and x(i) = x(i ′ ) mod t, then x(i)− x(i ′ ) is a multiple of t, say x(i)− x(i ′ )=qt for some value q. Then x(j − 1 qt+i ′ )=x(j − 1 qt)+x(i ′ )=jj − 1 qt+x(i ′ )=qt+x(i ′ )=x(i)− x(i ′ )+x(i ′ )=x(i). Therefore, because x is a bijection, j − 1 qt + i ′ = i mod n and so i− i ′ is a multiple of t. However, we assumed that both i and i ′ were positive integers smaller than t so their difference is only a multiple of t if their difference is 0. Thus, x mod t actually defines a permutation in S t− 1 . Specifically, for each x, there exists an associated remainder permutation σ x ∈ S t− 1 defined by σ x (i) := x(i) mod t for all i < t and a value j which is coprime to n t such that x(t)=jt. Furthermore, x(i)=tu i +σ x (i) for all 1≤ i≤ t− 1. Now, if we let x − 1 (i)=tl i +σ x − 1(i), then we obtain i=x(x − 1 (i))=x(tl i +σ x − 1(i)) =l i x(t)+x(σ x − 1(i)) =l i jt+tu σ x − 1 (i) +σ x (σ x − 1(i)) Now, note that t divides l i jt+tu σ x − 1 (i) , however, σ x ,σ x − 1 ∈S t− 1 and so they can only output values strictly smaller than t. This gives us two relations, first, σ x (σ x − 1(i)) = i for 39 all i, and so σ − 1 x = σ x − 1 which is very desirable, but also l i jt =− tu σ − 1 x (i) for all i, which we can rewrite to obtain l i =− j − 1 u σ − 1 x (i) mod n t for all i=1,...,t− 1. Corollary 6.1.5. Let x∈ S n− 1 have stabilizer F x =⟨a t ⟩. Then there exists integers j,u i for 1≤ i≤ t− 1 and remainder permutation σ x ∈S t− 1 as from Lemma 6.1.4 such that x(i)=u i t+σ x (i) mod n for all 1≤ i≤ t− 1 x(qt)=qjt mod n for all 1≤ q≤ n t x(qt+i)=qx(t)+x(i) mod n Then, if x is an involution, σ x is as well and u i =− ju σ x(i) for all 1≤ i≤ t− 1. Conversely, every choice of j coprime to n t , involution σ ∈ S t− 1 , and integers u i satis- fying u i =− ju σ (i) will generate an involution x∈S n− 1 which is stabilized by a t . Proof. Lemma 6.1.4 gives that σ x − 1 =σ − 1 x and so if x 2 =1 then σ 2 x =1. Note because x − 1 = x, j − 1 = j, and l i = u i for all i. Therefore, u i = − j − 1 u σ − 1 x (i) = − ju σ x(i) mod n t for all i. This says that for every fixed point of σ x , we require that u i (j+1)=0 mod n t . The converse follows by direct computation. Notethatknowingσ x isaninvolutiondoes notguaranteethatxisaswell. Forexample, if t=3, then every σ ∈S 2 has order 2, but certainly x∈S n− 1 for arbitrary n need not be. It must be stressed here that the remainder permutation σ x plays a vital role in under- standing (and thus counting) involutions x with certain properties. While we do not explore it in this work, we believe that irreps in J n are in some way connected to and possibly even extended from irreps in J t (for t dividing n) and that remainder permutations are the key to understanding this connection. 2 Counting Involutions with Certain Properties Here we focus on counting formulas for involutions in x ∈ S n− 1 with stabilizer F x = ⟨a t ⟩ and certain additional properties. 40 Table 6.1: A table of sets and values which are used heavily in the following statements. Number Set or Value (1) E n/t = n 1≤ j≤ n t j 2 =1 mod n t o (2) K j,n/t = n 1≤ u≤ n t u(j+1)=0 mod n t o (3) α j,n/t =gcd j+1, n t =|K j,n/t | (4) P j,n/t = n q(j− 1) mod n t 1≤ q≤ n t o (5) P (c) j,n/t = n 1≤ u≤ n t u̸∈P j,n/t o \ K j,n/t (6) δ P c =        |P (c) j,n/t | if P (c) j,n/t ̸=∅ 1 otherwise (7) K j ′ ,jσ x ,n,t,s = 1≤ u≤ n t u t s +mi (1+j ′ ) = 0 mod n s for some mi∈P jσ x ,t/s (8) K (c) j ′ ,jσ x ,n,t,s = 1≤ u≤ n t | u t s +mi (1+j ′ ) = 0 mod n s for some mi∈P (c) jσ x ,t/s (9) δ K c =        |K (c) j ′ ,jσ x ,n,t,s | if K (c) j ′ ,jσ x ,n,t,s ̸=∅ 1 otherwise (10) E jσ x ,n,t,s = j σ x +m t s 1≤ m≤ n t \ E n/s (11) β j,n/t =gcd j− 1, n t (12) δ j,n/t,r =        1 if β j,n/t |r and r≤ tβ j,n/t 0 otherwise (13) m r,j,n/t = r β j,n/t (14) K ′ j,n/2 = u|(u+1)(j+1)=0 mod n 2 (15) δ u,j,n/2 =        1 if j̸=2u+1 mod n 2 0 otherwise (16) δ 0 i,u,n/2 =        1 if − i(u+1)=0 mod n 2 0 otherwise (17) δ ̸=0 i,u,n/2 =        1 if − i(u+1)̸=0 mod n 2 0 otherwise 41 Proposition 6.2.1. Let T n/t be the set of involutions x∈ S n− 1 such that F x =⟨a t ⟩. Let E n/t be as from (1) and α j,n/t be as from (3) in Table 6.1 (Table 6.1). Then |T n/1 |=|E n/1 | (2.1) |T n/t |= X j∈E n/t ⌊ t− 1 2 ⌋ X l=0 (α j,n/t ) t− 1− 2l n t l (t− 1)! (t− 1− 2l)!2 l l! − X s|t s̸=t |T n/s | (2.2) |T n/n |= n− 1 2 X l=0 (n− 1)! (n− 1− 2l)!2 l l! − X s|n s̸=n |T n/s | (2.3) Proof. Let x∈S n− 1 be an involution with stabilizer F x =⟨a t ⟩ and let E n/t := n 1≤ j≤ n|j 2 =1 mod n t o . From Corollary 6.1.5, let σ x ∈ S t− 1 , write x(i) = tu i +σ x (i) for all 1 ≤ i ≤ t− 1 and x(t)=jt for some j∈E n/t which determines x completely. Therefore, the number of x satisfying x 2 =1 can be counted by fixing a value j∈E n/t , then for each involution σ ∈S t− 1 , we count the number of fixed points of σ which requires counting the number of transpositions present in the disjoint cycle decomposition of σ. Note, there are (t− 1)! (t− 1− 2l)!2 l l! permutations σ ∈S t− 1 with σ 2 =1 and σ being a product of l disjoint cycles. This is because we choose any two integers between 1 and t− 1 to put in a transposition (so t− 1 2 = (t− 1)(t− 2) 2 , then we choose any two of the remaining t− 3 integers to pair up in a second transposition so (t− 3)(t− 4) 2 ), and we repeat this l times which gives (t− 1)(t− 2)(t− 3)··· (t− 1− 2l) 2 l l! where we must divide by l! because the order in which we obtain each transposition does not matter, that is, (1 2)(3 4) = (3 4)(1 2) and there are l! ways to re-order the product of transpositions. Now, if x has remainder permutation σ , then from Lemma 6.1.4, we know that u i = − ju σ (i) mod n t andsoeachfixedpointof σ (ofwhichtherearet− 1− 2l)forcesu i (j+1)=0 mod n t which contributes gcd(j + 1, n t ) possible choices for u i where σ (i) = i (see last paragraph of the proof of Lemma 6.1.4). Furthermore, each transposition (i i ′ ) in σ which pairs u i with u i ′ and so contributes n t choices for u i (which then determines u i ′). 42 Let α j,n/t :=gcd j+1, n t . Therefore, for a fixed j, and fixed σ comprised of l disjoint transpositions, there are (α j,n/t ) t− 1− 2l n t l choices for the u i and so if we multiply this by the number of such σ , which is (t− 1)! (t− 1− 2l)!2 l l! and then sum over all j ∈ E n/t t, and subtract those we have already counted, we obtain our result |T n/t |= X j∈E n/t |{z} choices for x(t) ⌊ t− 1 2 ⌋ X l=0 |{z} # of fixed points of σ x (α j,n/t ) t− 1− 2l | {z } choices for x(i) when σ x(i)=i n t l |{z} choices for x(i) when σ x(i)̸=i (t− 1)! (t− 1− 2l)!2 l l! | {z } choice for σ x − X s|t s̸=t |T n/s | | {z } remove over count In the case where t=1, then l =0 and all powers are 0 so we are simply summing 1 for every j∈E n/1 . Of course, if t=n, all sets trivialize and so we simply count all involutions in S n− 1 and then remove those we have already counted. Again, as with Proposition 6.1.1, explicitly writing out the set T n/t is done recursively by using the formula given. At last, we can now verify the statements made at the end of the discussion under Theorem 5.1.2 that if n = p r for p and odd prime and r ∈ N, then there exists only one nontrivial involution x ∈ S n− 1 with F x = F. Since E p r /1 = {1,− 1}, we have only two permutations in T p r /1 = (1),(1 p r − 1)(2 p r − 2)··· p r − 1 2 p r +1 2 . The second permutation x in T p r ,1 is constructed by writing x(i) =− i for all i. For a further discussion of using Proposition 6.2.1 to generate T n/t , see Appendix A Appendix 3. Lemma 6.2.2. Let x ∈ S n− 1 be an involution with F x = ⟨a t ⟩. Let j ∈ E n/t as from Table 6.1 (Table 6.1) and write x(t) = jt and for all 1≤ i < t, and x(i) = u i t+σ x (i) for some involution σ x ∈S t− 1 and constants 1≤ u i ≤ n t . Let β j,n/t =gcd j− 1, n t =gcd 1− j, n t . Then, x(qt+i)=qt+i for β j,n/t different values q if and only if σ x (i)=i and u i =q(1− j) mod n t for some q. 43 Proof. =⇒ Let x(qt+i)=qt+i be a fixed point of x. Then this says that x(qt+i)=qx(t)+x(i)=qjt+u i t+σ x (i)=qt+i. Because σ x (i)<t and i<t, this forces σ x (i)=i. Thus, subtracting we obtain that u i t=qt− qjt =⇒ u i =q(1− j) mod n t . ⇐= Let σ x (i) = i and u i = q(1− j) mod n t for some q. Now, since there exists a q for which the equation u i = q(1− j) mod n t has a solution, there exists a total of β j,n/t solutions and so there are β j,n/t possible values q for which x(qt+i)=qjt+u i t+σ x (i)=qjt+q(1− j)t+i=qt+i. Proposition 6.2.3. Let R n/t,r be the set of involutions x∈ S n− 1 such that F x =⟨a t ⟩ and x has exactly r fixed points (including x(n)=n as one of the fixed points). Let E n/t , β j,n/t , m r,j,n/t , δ j,n/t,r , P j,n/t , δ P c as from Table 6.1. Let δ r be the Kronecker-Delta step function. Then |R n/1,r |= X j∈E n/1 δ r (β j,n/1 ) (2.4) |R n/t,r |= X j∈E n/t ⌊ t− m r,j,n/t 2 ⌋ X l=0 δ j,n/t,r δ l j,n/t,r t− 1− 2l m r,j,n/t − 1 |P j,n/t | m r,j,n/t − 1 (δ P c ) t− 2l− m r,j,n/t n t l (t− 1)! (t− 1− 2l)!2 l l! − X s|t s̸=t |R n/s,r | (2.5) |R n/n,r |= (n− 1)! (r− 1)!2 n− r 2 n− r 2 ! − X s|n s̸=n |R n/s,r | (2.6) |R n/t,r |=0 if n− r is not even. Proof. Let x∈ S n− 1 be an involution with stabilizer F x =⟨a t ⟩. For some σ x ∈ S t− 1 , write x(i)=tu i +σ x (i) for all 1≤ i≤ t− 1 and x(t)=jt for some j∈E n/t which determines x completely as from Corollary 6.1.5. Then, assume x has r fixed points (including x(n)=n as a fixed point). 44 Since x(qt) = jqt = qt if and only if q(j − 1) = 0 mod n t , we immediately get β j,n/t :=gcd(j− 1, n t ) fixed points for x from the given choice of j. Additionally, from Lemma 6.2.2, we know that for a fixed 1 ≤ i < t, x(qt+i) = qt+i for β j,n/t different values of q if and only if σ x (i)=i and u i =q(1− j) mod n t . Thus, the total number of fixed points of x is a multiple of β j,n/t and so β j,n/t must divide r. To ensure the existence of an x with the given choice of j and r, the value δ j,n/t,r must be included in our sum where δ j,n/t,r :=        1 if β j,n/t |r and r≤ tβ j,n/t 0 otherwise . Furthermore,lettingm r,j,n/t := r β j,n/t ,theratiominusone,m r,j,n/t − 1tellsushowmany u i for 1≤ i<t must be of the form q(1− j) for some q. Let P j,n/t := n q(1− j) mod n t |1≤ q≤ n t o . Then m r,j,n/t − 1 of the u i come from P j,n/t , and t− 1− (m r,j,n/t − 1) = t− m r,j,n/t of the remaining u i come from its compliment P c j,n/t := n u|u̸=q(1− j) mod n t for all q o . However, recall that if σ x (i)=i, then u i (j+1)=0 mod n t is required by Corollary 6.1.5. Of course, if u i ∈ P j,n/t , then u i = q(j − 1) for some q and so certainly u i (j + 1) = q(j− 1)(j+1)=q(j 2 − 1)=0 since j 2 =1. However, to guarantee this for u i ∈P c j,n/t , we must actually require that u i ∈P (c) j,n/t :=P c j,n/t ∩K j,n/t where K j,n/t := n 1≤ u≤ n t |u(j+1)=0 mod n t o . In summary: 1. If σ x (i)=i, then either: (a) u i ∈P j,n/t and so contributes β j,n/t fixed points to x (b) or u i ∈P (c) j,n/t and so contributes 0 fixed points to x 2. If σ x (i)̸=i then x(qt+i)̸=qt+i for any q and so we can allow u i to take any value between 1 and n t and this will not add any fixed points to x. 45 Let t− 1− 2l be the number of fixed points of σ x for some l. Then, since m r,j,n/t − 1 of the u i come from P j,n/t , σ x must have at least m r,j,n/t − 1 fixed points which means t− 1− 2l≥ m r,j,n/t − 1 and so t− m r,j,n/t ≥ 2l so finally l≤ t− m r,j,n/t 2 . Additionally, for the remaining t− 1− 2l− (m r,j,n/t − 1)=t− 2l− m r,j,n/t fixed points of σ x , u i must be taken from P (c) j,n/t , however, there is an edge case we must address here. Assume P (c) j,n/t =∅. Then either P c j,n/t =∅ (which is possible) or K j,n/t ⊂ P j,n/t . In the first case, we get P j,n/t ={1,2,...,n/t} and since P j,n/t ⊂ K j,n/t , we get K j,n/t =P j,n/t . In the second case, K j,n/t ⊂ P j,n/t and so again K j,n/t = P j,n/t . In either case, if i is a fixed point of σ x , and so σ x (i) = i, then because u i ∈ K j,n/t = P j,n/t by Corollary 6.1.5, x has β j,n/t fixed points of the form qt+i for some q. Namely, if P (c) j,n/t =∅, every fixed point of σ x contributes exactly β j,n/t fixed points to x and so the number of fixed points of σ x is equal to m j,n/t − 1. That is t− 1− 2l =m j,n/t − 1 and so t− m j,n/t =2l. All this is to say that if P (c) j,n/t =∅, then we must have t− m j,n/t =2l so we introduce δ l j,n/t,r =        0 if|P (c) j,n/t |=∅ and 2l̸=t− m j,n/t 1 otherwise However, this condition is messy to write down as we do not want to be stuck with a 0 0 indeterminate in our sum so to avoid this, the set P (c) j,n/t is never used. Instead, we include the messy conditional δ P c :=        |P (c) j,n/t | if P (c) j,n/t ̸=∅ 1 otherwise . Now, we are at last ready to fully describe all involutions with stabilizer F x =⟨a t ⟩ with and r fixed points. In the base case, if t = 1, then x(i) = ij = i if and only if i(j− 1) = 0 mod n. So x has r fixed points if and only if the number of solutions to this equation is r. Since i = n is always a solutions, the number of solutions to this equation is β j,n/1 =gcd(j− 1,n). 46 Let δ r be the step function so δ r (u)=        1 if r =u 0 otherwise Then |R n/1,r |= X j∈E n/1 δ r (β j,n/1 ). Finally, |R n/t,r |= X j∈E n/t |{z} choices for x(t) ⌊ t− m r,j,n/t 2 ⌋ X l=0 | {z } # of fixed points of σ x δ j,n/t,r | {z } x exists δ l j,n/t,r | {z } x matches σ x t− 1− 2l m r,j,n/t − 1 | {z } choose u i in P j,n/t |P j,n/t | m r,j,n/t − 1 | {z } σ x(i)=i and u i =q(1− j) (δ P c ) t− 2l− m r,j,n/t | {z } σ x(i)=i and u i ̸=q(1− j) n t l |{z} σ x(i)̸=i (t− 1)! (t− 1− 2l)!2 l l! | {z } choices for σ x − X s|t s̸=t R n/s,r | {z } remove over count Note that this agrees with our base case since if t = 1 we know that 1 ≥ m r,j,n/t and that m r,j,n/t is an integer so m r,j,n/t =1. Thus, necessarily r =β j,n/t (or of course no such x exists) and so l =0. So the sum becomes X j∈E n/1 δ j,n/t,r = X j∈E n/1 δ r (β j,n/t ) as desired. Furthermore, If t = n, all sets trivialize and so we simply count all involutions with r fixed points in S n− 1 and then remove those we have already counted. Note also that if n− r is not even, then x cannot be an involution in S n− 1 with r fixed points, and so we say that|R n/t,r |=0 if n− r is not even. Lemma 6.2.4. Either P j,n/t = K j,n/t and α j,n/t β j,n/t = n t or |P j,n/t | = 1 2 |K j,n/t | =|P (c) j,n/t | and α j,n/t β j,n/t =2 n t Proof. Recall that β j,n/t := gcd 1− j, n t and P j,n/t := {q(1− j) mod n t }. We claim |P j,n/t |= n tβ j,n/t . Thisisbecauseq(1− j)=q ′ (1− j)ifandonlyif(q− q ′ )(1− j)=0 mod n t if and only if q =q ′ mod n tβ j,n/t which proves the claim. 47 Recall now that P j,n/t ⊂ K J,n/t ⊂{ 1,2,..., n t } is always true and so the results follow by checking the cases where P (c) j,n/t =∅ and P (c) j,n/t ̸=∅ separately and using basic properties of the gcd. Remark 6.2.5. By definition P n r=1 |R n/t,r | = |T n/t | must be true, however, using some careful computation it can be shown that both equations from Proposition 6.2.1 and Propo- sition 6.2.3 satisfy this relation. 3 Prerequisites for Counting Indicators Now, lets say we want to count the number of x∈S n− 1 which have indicators of a certain value. Note that if x 2 = 1, then x = x − 1 ∈O x and so by Proposition 5.1.5, all indicators are nonnegative. Now, what determines whether the indicator of x is zero or not? Well, ν ( ˆ V)=1 if ζ i x(t)+t t n =1 and 0 otherwise. Therefore, knowing the value of x(t) tells us exactly which (and so how many) i exist for which ν ( ˆ V)̸= 0. Now, assume that t is odd. Then O x is closed under inversion if and only if it contains an element y of order 2. Since O y = O x , we may relabel so as to only consider orbitsO x where x is an involution. And so, in the case where t is odd, the strategy is this: 1. x − 1 ∈O x if and only ifO x contains an element of order 2. 2. The number of involutions (excluding x itself) inO x is given by the number of fixed points of σ x ∈S t− 1 which generates x (by Lemma 4.3.5 Item 2). 3. The number of representations of J n induced form x which have positive indicator is given by the number of values i for which ζ i x(t)+t t n =1 (by Theorem 5.1.2). Namely,ifwewanttocountthenonzeroindictorsofirrepsofJ n ofafixedodddimension t, we must be able to count the number of orbits which contain exactly r involutions. Proposition 6.3.1. Let X n/t,r be the set of involutions x ∈ S n− 1 with F x = ⟨a t ⟩, and such that O x has r̸= 0 involutions. Let T n/1 be as from Proposition 6.2.1, R n/t,r as from 48 Proposition 6.2.3, and we pull the entries (1), (2), (4), (6), (7), (9), (10), (11), (12), (13) from Table 6.1 (Table 6.1). Then |X n/1,r |=|T n/1 |=|E n/1 | (3.1) |X n/t,r |= X j∈E n/t (α j,n/t ) r− 1 n t t− r 2 (t− 1)! (r− 1)!2 t− r 2 t− r 2 ! − X s|t s̸=t |C n,t,s,r | (3.2) |X n/n,r |= (n− 1)! (r− 1)!2 n− r 2 n− r 2 ! − X s|n s̸=n |C n,n,s,r | (3.3) |X n/t,r |=0 if t− r is negative or odd where |C n,t,1,r |= X jσ x ∈E t/1 δ r (β jσ x ,t/1 )|E jσ x ,n,t,1 | (3.4) |C n,t,s,r |= X jσ x ∈E t/s X j ′ ∈E jσ x ,n,t,s ⌊ s− m r,jσ x ,t/s 2 ⌋ X l=0 δ jσ x ,t/s,r δ l jσ x ,t/s,r s− 1− 2l m r,jσ x ,t/s − 1 |P jσ x ,t/s | m r,jσ x ,t/s − 1 (δ P c ) s− 2l− m r,jσ x ,t/s (s− 1)! (s− 1− 2l)!2 l l! |K j ′ ,jσ x ,n,t,s | m r,jσ x ,t/s − 1 (δ K c ) s− 2l− m r,jσ x ,t/s n s l − X p|s p̸=d |C n,t,p,r | (3.5) |C n,n,s,r |=|R n/s,r | (3.6) Proof. Once again, we count. Assume x∈ S n− 1 is an involution with F x =⟨a t ⟩ such that O x contains exactly r involutions. Our goal is to fully describe the conditions necessary to construct x and then count the possible ways to achieve those conditions. We know that there exists integers u i and permutation σ x ∈ S t− 1 such that for all 1 ≤ i ≤ t− 1, x(i) = tu i + σ x (i). Furthermore, we know that σ x has exactly r fixed points (including σ x (t) = t). This is because Lemma 4.3.5 Item 2 tells us that if we write O x = {x = y t ,y 1 ,y 2 ,...,y t− 1 }, then y i ∈ O x has order 2 if and only if i = x(i) mod t = σ x (i). SinceO x is assume to have r involutions (including x), σ x must then have r fixed points. Now, we can adapt the formulas from Proposition 6.2.1 here, however, there will be a few key differences. Let j ∈ E n/t be fixed. Then recall that the number of involutions 49 σ ∈S t− 1 which are a product of l disjoint cycles is given by (t− 1)! (t− 1− 2l)!2 l l! . In this case, σ has t− 1− 2l fixed points. In our case we want t− 1− 2l = r− 1 (we now exclude σ x (t) as a fixed point since we are viewing σ x as strictly a permutation in S t− 1 ) and so we obtain that l = t− r 2 . Note that t− r must be even (since this is the number of permutations y∈O x which are not involutions) this value is an integer. Finally, we obtain that there are (t− 1)! (r− 1)!2 t− r 2 ( t− r 2 )! involutions σ ∈S t− 1 with r− 1 fixed points. Thus, there should be X j∈E n/t (α j,n/t ) r− 1 n t t− r 2 (t− 1)! (r− 1)!2 t− r 2 t− r 2 ! involutions x ∈ S n− 1 with stabilizer ⟨a t ⟩ which have r fixed points mod t. This should seem familiar as it is the same formula as from Proposition 6.2.1 with l fixed. As with Proposition 6.2.1, the term α j,n/t dictates that if σ x (i)=i, then u i ∈K j,n/t , while the term n t dictates that if σ x (i)=i ′ , then u i =u i ′. However, we have–unfortunately–over counted. What this value really gives us, is the number of involutions x∈ S n− 1 which have r fixed points mod t and which are stabilized by a t . However, their actual stabilizer set could be larger than ⟨a t ⟩. To compensate for this, we must remove those permutations which have smaller stabi- lizer. However, we cannot just do this recursively. Note that we are counting the number of fixed points of x modulo t, which we will not do if we simply replace every t with an s. Rather, we must work harder here. Certainly, if x is actually stabilized by a s for some s|t, then there exists a τ x ∈ S s− 1 such that x(i) = l i s+τ x (i) for all i < s. This is to say that we have two equivalent ways of writing x(i). Namely, for all i

Abstract (if available)

Abstract The semisimple bismash product Hopf algebra Jn = kSn−1 #kCn for an algebraically closed field k is constructed using the matched pair actions of Cn and Sn−1 on each other. In this work, we reinterpret these actions and use an understanding of the involutions of Sn−1 to derive a new Froebnius-Schur indicator formula for irreps of Jn and show that for n odd, all indicators of Jn are nonnegative. We also derive a variety of counting formulas including Theorem 7.2.2 which fully describes the indicators of all 2-dimensional irreps of Jn and Theorem 7.1.2 which fully describes the indicators of all odd-dimensional irreps of Jn and use these formulas to show that nonzero indicators become rare for large n

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Creator Orlinsky, Kayla (author)

Core Title An indicator formula for a semi-simple Hopf algebra

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Degree Conferral Date 2023-05

Publication Date 04/18/2023

Defense Date 04/18/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag algebra,aymmetric group,Bismash product algebra,factorizable group,Forbenius-Schur indicator,group theory,Hopf algebra,OAI-PMH Harvest,representation theory,semisimple algebra

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Montgomery, Susan (committee chair), Guralnick, Robert (committee member), (Pilch, Krzysztof)

Creator Email kaylaorlinsky@gmail.com,korlinsk@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113038288

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