arXiv:1912.08576v2 [math.RT] 21 Mar 2020 h groups irreduc the dimensional, (finite between available identities lution groups yeothda group hyperoctahedral lsrt osdrirdcbeuioetrepresentations unipotent irreducible consider to closer fir identities, of character tions Shintani as known classically are stecnrlzro xdpitfe involution free point fixed a of centralizer the as we h reuil ersnain ftegroups the of representations irreducible the tween efix We ymti group symmetric ymti group symmetric HRCE EAINHPBTENSMERCGOPAND GROUP SYMMETRIC BETWEEN RELATIONSHIP CHARACTER A .Introduction 1. Let 1991 Proposition of Proof A. Appendix theorem References The Littlewood of theorem 6. of A representations between correspondence of 5. Example representations of 4. Basechange mapping Norm 3. The 2. RN L FRANK g G ahmtc ujc Classification. Subject Mathematics srltdt h hrce hoyo h xdsbru fthe of subgroup fixed the of theory character the automorp to diagram related with is groups reductive of theory character hto h yeothda group hyperoctahedral the of that SO A w GL → BSTRACT 0 eete h ymti group symmetric the either be 2 (1 = n GL n t ( +1 BC N IEDAPAA,WT NAPNI YAVN AYYER ARVIND BY APPENDIX AN WITH PRASAD, DIPENDRA AND UBECK g ¨ F − 2 ( q 1 n C ) , ih(nt iesoa,irdcbe leri)repres algebraic) irreducible, dimensional, (finite with , ( erlt hrce hoyo h ymti groups symmetric the of theory character relate We . − and ) S S C or , 2 2 1)(2 ) n n or , +1 GL satn nteset the on acting as Sp B , satn nteset the on acting as n − n GL 2 YEOTHDA GROUP HYPEROCTAHEDRAL ( n ( = 2) F ( C 2 q · · · n d Z ) +1 ) o hc erfrt [ to refer we which for , f [ cf. , / ( ( 2) n, C .I 1. n B ) − S ⋊ Sh n hc r efda,ie,ivratudrteinvo- the under invariant i.e., self-dual, are which rmr 17;Scnay22E55. Secondary 11F70; Primary n C NTRODUCTION 5.4 ( = ) ,atog o h aea ad twudb much be would it hand, at case the for although ], ONTENTS n h ae rvsacaatrrltosi be- relationship character a proves paper The . S itn aual inside naturally sitting , Z yAvn Ayyer Arvind by {± 2 / n 1 { 2) 0 rtesmercgroup symmetric the or 1 n , , ± S ⋊ ± 1 2 w G , n , ± 0 spr fteepcainta the that expectation the of part as , · · · and in 2 be leri)rpeettosof representations algebraic) ible, , KLP fsay of , · · · tosre ewe representa- between observed st S ± B imadterWy groups, Weyl their and hism iga automorphism. diagram 2 n n n , nti ae,w aethe take we paper, this In . .Teecaatridentities character These ]. } lsl eae ocharacter to related closely , ± S fcardinality of U n S 2 2 } n 2 n G n ( and fcardinality of F and ( B q ) S n S S orsodn to corresponding 2 2 2 ⊂ ac 4 2020 24, March nain fthe of entations n n n +1 +1 +1 S and , 2 2 with n n ⊂ n the and , 2 S n H 2 1 + n +1 the 20 19 17 3 1 9 7 4 ) . 2 FRANKLUBECK¨ AND DIPENDRA PRASAD an irreducible representation of the Bn and the associated basechanged rep- resentation of GL2n(Fq2 ) associated to a representation of the Weyl group S2n, and which are related by a basechange character identity, cf. [Ka], [D]. Observe that in all the cases above, the group G comes equipped with an automorphism, call it j, of finite order (such as conjugation by w0 for symmetric groups), and H is either the subgroup of fixed points of this automorphism, or is closely related to the subgroup of fixed points (through a dual group construction as in [KLP]), and in the theory of basechange, it is not so much the representation theory of the group G which is important, but rather it is the representation theory of the group G ⋊ hji. Basic to character identities in all the cases referred above, are two maps (correspondence in general): (1) A correspondence BC between irreducible representations of H and irreducible representations of G, to be called basechange of representations of H to represen- tations of G, which in all the cases listed above is an injective map from irreducible representations of H to irreducible representations of G (but not surjective except in some trivial cases). (2) A correspondence Nm between (certain) j-twisted-conjugacy classes in G (two elements g1,g2 in G are said to be j-twisted-conjugate if there exists g ∈ G such −1 −1 that g1 = gg2jg j ) and (certain) conjugacy classes in H, to be called the norm mapping from G to H, and denoted Nm : G → H. In the various examples mentioned above, it is the map BC between irreducible repre- sentations of H and irreducible representations of G that is non-trivial to define or con- struct, whereas the norm map Nm : G → H is usually straightforward to define, and is defined on all of G. However, in the case at hand of symmetric groups, norm mapping is also not so trivial to define especially as it is not defined on all of G. The main theorem of this work, Theorem 6.1, proves the following character relation:

Θ(BC(π))(w)= ǫ(BC(π))Θ(π)(Nmw), for any irreducible representation π of Bn, and w ∈ G which is a product of even cycles with at most one fixed point (where ǫ(BC(π)) = ±1). In particular, taking w = w0, we have

Θ(BC(π))(w0)= ǫ(BC(π))Θ(π)(1), which was at the origin of this work. After the completion of this work, Prof. G. Lusztig informed the authors that this special case occurs on page 110 of his paper [Lu]. The above theorem was arrived at by computations done via the GAP software [GAP], and inspired by the hope that basechange character identities available in many situations involving reductive groups, also have an analogue for Weyl groups of these algebraic groups. We eventually found that it is a simple consequence of Theorem 5.7 due to Lit- tlewood in [Li1] for even symmetric groups. Along the way, we have given a complete proof of this theorem of Littlewood both for S2n as well as S2n+1. It is surprising that the desired character identity relating symmetric groups and hyperoctahedral group is proved via Frobenius character formula for symmetric groups which is a form of Schur-Weyl du- ality by a “factorization” of the character formula (i.e., Schur polynomials) for irreducible representations of GL2n(C) on special elements discovered by the second author in [P], which is already there in [Li1] written in 1940! A CHARACTER RELATIONSHIP BETWEEN AND HYPEROCTAHEDRAL GROUP 3

2. THE NORMMAPPING The aim of this section is to define a map from certain conjugacy classes in G to con- jugacy classes in H where G is either the symmetric group S2n or the symmetric group n S2n+1, and H is the hyperoctahedral group Bn =(Z/2) ⋊ Sn, sitting naturally inside G as the centralizer of the fixed point free involution w0 = (1, −1)(2, −2) ··· (n, −n). We will call this map the norm mapping and denote it as Nm : G → H even though it is not defined on all of G. Dealing with both the groups S2n and S2n+1 at the same time has some notational issues, so we first deal with G = S2n, and then come back to G = S2n+1 at the end. We n let Bn =(Z/2) ⋊ Sn be defined more concretely as: n Bn =(Z/2) ⋊ Sn = {h ∈ Aut{±1, ±2 ··· , ±n}|h(−i)= −h(i), ∀i}, with Sn ⊂ Bn be the subgroup of permutations σ of {1, 2 ··· , n} extended to the points {−1,..., −n} by σ(−i)= −σ(i). We begin by recalling that conjugacy classes in the symmetric group Sm are in bijective correspondence with partitions p of m, which are collection of nonzero {p1 ≥ p2 ≥···≥ pr ≥ 1}. Conjugacy classes in Bn are also well-understood beginning with the observation that there are two Bn-conjugacy classes among elements of Bn which project to an n-cycle, n n say sn = (123 ··· n), in Sn. If a · sn ∈ (Z/2) ⋊ Sn with a ∈ (Z/2) , then the Bn- conjugacy class of a · sn is determined by tr(a) ∈ Z/2 where a → tr(a) is the homo- morphism tr : (Z/2)n → Z/2 which is adding-up of the co-ordinates of a. Thus one may call, the two Bn-conjugacy classes among elements of Bn which project to an n- cycle in Sn as positive n-cycle and negative n-cycle depending on whether tr(a)=0 or tr(a)=1. These two conjugacy classes can also be differentiated by the order of their elements which is n for a positive n-cycle, and is 2n for a negative n-cycle. n It follows that conjugacy classes in Bn =(Z/2) ⋊ Sn are in bijective correspondence with (ordered pairs of) partitions {p0, p1} of n (to be called a bi-partition, or a pair of 0 1 partitions of n), i.e., with |p0| + |p1| = n, with positive pi cycles and negative pi cycles 0 1 where pi (resp. pi ) are the parts of the partition p0 (resp. p1). The natural map from the set of conjugacy classes in Bn to the set of conjugacy classes in S2n takes the conjugacy class corresponding to a bi-partition {p0, p1} of n to the con- jugacy class in S2n corresponding to the partition {p0, p0, 2p1} of 2n where 2p1 stands for the partition in which each component of p1 is multiplied by 2. As a consequence of the above description of conjugacy classes in Bn ⊂ S2n, observe that the natural mapping from the set of conjugacy classes in Bn to the set of conjugacy classes in S2n is not one-to-one. For example, the conjugacy class of the fixed point free involution w0 in S2n intersects with exactly with [n/2]+1 conjugacy classes in Bn (where [x] denotes the integral part of a real number x) represented by the bi-partitions + − + − {pd ,pd } where pd is the partition {2, 2, ··· , 2} of 2d ≤ n, and pd is the trivial partition {1, 1, ··· , 1} of (n − 2d). Definition 2.1. (Norm mapping) The norm mapping from the set of conjugacy classes n in S2n to the set of conjugacy classes in Bn = (Z/2) ⋊ Sn is defined on those conju- gacy classes g in S2n represented only by even cycles {2p1 ≥ 2p2 ≥ ··· ≥ 2pr ≥ 2} (these conjugacy classes in particular have no fixed points under the action of S2n on 4 FRANKLUBECK¨ AND DIPENDRA PRASAD

{±1, ±2, ··· , ±n}) which are mapped to Nm(g) = h, a conjugacy class in Bn = n (Z/2) ⋊ Sn corresponding to the bi-partition p = (p0, p1) of n having only positive cycles, i.e., p1 is empty, and p0 = {p1 ≥ p2 ≥···≥ pr ≥ 1}. n The norm mapping from S2n+1 to Bn = (Z/2) ⋊ Sn is defined on the set of those conjugacy classes g in S2n+1 which come from S2n, and for them the norm mapping is the one just defined for S2n above. Remark 2.2. There is another way to define the norm mapping which relates to more conventional way norm mappings are defined in questions about basechange character identities (say, relating twisted characters for G(Fqd ) to characters of G(Fq)), which we briefly discuss now. Recall that there is a natural mapping from representations π of S2n ′ to representations π of the semidirect product group S2n ⋊ hw0i in which the underlying ′ ′ vector space for π is the same as that for π, and for which w0 ∈hw0i operates on π as w0 does on π. Under this identification of representations of S2n with one of S2n ⋊ hw0i, the ′ character of π at an element g ⋊ w0 is the same as the character of π at the element gw0. ′ In the basechange character identity, one relates the character of π at an element g ⋊ w0 (called the twisted character of π′ at the element g) to the character of a representation of 2 the subgroup Bn at the element of Bn which is essentially g · (w0gw0)=(gw0) . Thus, in terms of π, we will be relating the character of π at gw0 to the character of a representation 2 of Bn, call it λ, at the element (gw0) which we do not know if it belongs to Bn, and even if it does, it may not define a unique conjugacy class in H. From the point of view of S2n, the element x = gw0 is of course any element, and then the question is if there is a good 2 supply of elements x ∈ S2n such that x ∈ Bn, and then tofind a suitable conjugacy class 2 in Bn containing x . The element x =1, and such ‘singular’ elements must be avoided if we are to have any hope of character identity of the representation π of S2n at x, with the representation λ of Bn at the element Nm(x). Since the of a 2d cycle is a product of two disjoint cycles of length d, it is clear by our earlier description of conjugacy classes in Bn that if x ∈ S2n is without fixed points for which all cycles are of even length, then x2 arises from a conjugacy class in Bn, although not a unique conjugacy class in Bn. However, there is a unique conjugacy class in Bn consisting only of positive cycles such that the intersection of the conjugacy 2 class of x in S2n with Bn is that conjugacy class in Bn, defining Nm(x) (on the set of elements x ∈ S2n which are product of disjoint even cycles without fixed points).

3. BASECHANGE OF REPRESENTATIONS

It is well-known that the irreducible representations of S2n are in natural bijective corre- spondence with partitions of 2n. We denote this correspondence by p → π(p). Similarly, n by Clifford theory, irreducible representations of Bn = (Z/2) ⋊ Sn are parametrized by an ordered pair of partitions {p0, p1} with |p0| + |p1| = n. Denote the corresponding irreducible representation of Bn by π(p0, p1). Before we describe the basechange map from irreducible representations of Bn to irre- ducible representations of S2n, we recall the notion of 2-core partitions, and 2-quotient of a partition. A partition is called a 2-core partition if none of the hook lengths in its Young diagram is a multiple of 2. (Same definition replacing 2 by any prime p defines a p-core partition.) It is easy to see that a 2-core partition exists for a number n if and only if the number n is A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP 5 a triangular number: d(d + 1) n = , 2 and in this case there is a unique 2-core partition which is the stair-case partition {d,d − 1,d − 2, ··· , 2, 1}.

Every partition p has a 2-core partition, call it c2(p) associated to it, and a pair of partitions (p0, p1) called the 2-quotient of p such that

|p| = |c2(p)| + 2(|p0| + |p1|). We briefly recall the definitions of 2-core and 2-quotient partitions of any partition, more generally with 2 replaced by any prime p, associated to a partition. The notion of a p-core and p-quotient of a partition is due to Littlewood, cf. [Li2], which arose there in his study of modular representations of the symmetric group. We will consider a partition interchangeably with the associated Young diagram which has the obvious notion of its rim. For a square in the Young diagram with hook length p, p any prime, one looks at its ‘arm’ and ‘leg’, and removes all the on the rim lying in-between those squares on the rim where the arm and the leg intersect the rim, and including these squares on the rim where the arm and leg intersect the rim. Iterating this process till there is no square in the Young diagram of hook length p, we come to a Young diagram called the p-core of the partition, or of its Young diagram; p-core of a partition is a p-core partition. Since in each step, one is removing p squares from the Young diagram, the size of the p-core is congruent to the size of the original Young diagram modulo p. Here is how one gets the 2-core of a Young diagram. Begin with the bottom most row of the diagram. If it has even length, remove it; if it has odd length, remove all boxes but one. If there are two consecutive rows (anywhere) which are both of even size or both of odd size, remove these two. At the end, you clearly get a Young diagram in which consecutive rows alternate being of even and odd size, with the minimal one of odd size. The 2-core of the partition is (d,d − 1, ··· , 1) where d is the number of rows left at the end. Now we define the p-quotient of a partition p which is an ordered collection of p parti- tions (p0, p1, ··· pp−1). Define β-numbers associated to a partition µ := {µ1 ≥ µ2 ≥ ··· ≥ µr} (we allow some of the µi to be zero) to be the collection of (now strictly decreasing) numbers

{µ1 +(r − 1),µ2 +(r − 2), ··· ,µr}. What needs to be kept in mind is that there is an equivalence relation on the set of parti- tions defined by declaring µ = µ1 ≥ µ2 ≥···≥ µr, and ν = ν1 ≥ ν2 ≥···≥ νs to be in the same equivalence class if and only if they are the same after adding a few 0’s at the tails. This equivalence gives rise to a similar equivalence on associated β-numbers (but which looks quite different); but the point is that from a sequence of β numbers (any strictly decreasing sequence of non-negative numbers) one can obtain the corresponding partition un-ambiguously. Coming back to the partition p = p1 ≥ p2 ≥ ··· ≥ pr, we now assume that r is divisible by p (adding zeroes at the end if necessary). For the associated β-numbers, β(p) = {p1 +(r − 1),p2 +(r − 2), ··· ,pr}, and for each i, 0 ≤ i ≤ p − 1, i let β (p) be those numbers in β(p) = {p1 +(r − 1),p2 +(r − 2), ··· ,pr} which are 6 FRANKLUBECK¨ AND DIPENDRA PRASAD congruent to i mod p. Define a new sequence of β-numbers by substracting i from each of the numbers in βi(p), and then dividing by p. These β numbers, call them βi(p), are associated to a partition pi, defining p-quotient of p, an ordered collection of p partitions (p0, p1, ··· pp−1). It can be seen that to any partition p, its p-core cp(p), and p-quotient (p0, p1, ··· pp−1), determine the partition p uniquely, and conversely, any data of a p-core and a p-quotient is associated to a partition p. In particular, for p = 2, and n any integer, the set of bi- partitions (p0, p1) of n is in bijective correspondence with partitions of 2n with empty 2-core, and 2-quotient, (p0, p1). We will construct the 2-quotient of a partition in some detail now to bring out a few points needed for our work. Let p = {p1 ≥ p2 ≥···≥ pm} be a partition of |p| with associated β-partition:

β(p) := {p1 +(m − 1) >p2 +(m − 2) > ··· >pm}. Consider the even and odd parts of this β-partition as:

0 β (p) := pi1 +(m − i1) >pi2 +(m − i2) > ··· >pik +(m − ik) 1 β (p) := pj1 +(m − j1) >pj2 +(m − j2) > ··· >pjℓ +(m − jℓ). Divide the numbers appearing in β0(p) and β1(p) − 1 by 2: β0(p) p +(m − i ) p +(m − i ) p +(m − i ) : i1 1 > i2 2 > ··· > ik k 2 2 2 2 β1(p) − 1 p +(m − j − 1) p +(m − j − 1) p +(m − j − 1) : j1 1 > j2 2 > ··· > jℓ ℓ . 2 2 2 2 These sequence of strictly decreasing numbers are β-numbers for the partitions: p +(m − i ) p +(m − i ) p +(m − i ) p : i1 1 − (k − 1) ≥ i2 2 − (k − 2) ≥···≥ ik k 0 2 2 2 p +(m − j − 1) p +(m − j − 1) p : j1 1 − (ℓ − 1) > ··· > jℓ ℓ . 1 2 2

It follows that (using m = k + ℓ, and the fact that the set of integers m − iα and m − jβ are exactly a permutation of the set of integers 0, 1, 2, ··· , m − 1),

|p| − 2(|p0| + |p1|) = 2(k − 1+ k − 2+ ··· +1+0)+2(ℓ − 1+ ℓ − 2+ ··· +1+0)

−[(m − i1)+(m − i2)+ ··· +(m − ik)]

−[(m − j1 − 1)+(m − j2 − 1) + ··· +(m − jℓ − 1)] m(m − 1) = k(k − 1) + ℓ(ℓ − 1) − + ℓ 2 (k − ℓ)(k − ℓ − 1) = . 2 As a consequence, we derive that

(1) |p| = 2(|p0| + |p1|) if and only if k = ℓ, or k = ℓ +1. Equivalently, the partition p has empty 2-core (thus necessarily with |p| even) if and only if k = ℓ, or k = ℓ+1; i.e., in the β numbers associated with p, either half of them are even and half of them are odd, or the even ones are one more than the odd ones. A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP 7

(2) |p| = 1+2(|p0| + |p1|) if and only if k +1= ℓ, or k = ℓ +2. Equivalently, the partition p has 2-core consisting of 1 (thus necessarily with |p| odd) if and only if in the β numbers associated with p, odd numbers are one more in cardinality than even numbers, or the even ones are two more than the odd ones. We summarize the above in the following proposition.

Proposition 3.1. A partition p has empty 2-core (thus necessarily with |p| even) if and only if k = ℓ, or k = ℓ +1, i.e., in the β numbers associated with p, either half of them are even and half of them are odd, or the even ones are one more than the odd ones. A partition p has 2-core consisting of 1 (thus necessarily with |p| odd) if and only if k +1= ℓ, or k = ℓ +2, i.e., in the β numbers associated with p, the odd numbers are one more in cardinality than the even numbers, or the even ones are two more than the odd ones. In particular, if |p| is even, and has even number of parts, then p has empty core if and only if k = ℓ, whereas if |p| is odd, and has odd number of parts, then p has 2-core 1 if and only if k +1= ℓ. (In our later analysis, we will always assume adding zeros at the end if necessary that if |p| is even, it has even number of parts whereas if |p| is odd, it has odd number of parts.)

Definition 3.2. (Basechange) Define the basechange of an irreducible representation π(p0, p1) of Bn to be the irreducible representation of S2n corresponding to the (unique) partition p such that the 2-core of p is empty, and the 2-quotient of p is (p0, p1), in par- ticular, we have |p| =2n = 2(|p0| + |p1|). Further, define the basechange of π(p0, p1) to S2n+1 to be the irreducible representation of S2n+1 corresponding to the (unique) parti- tion p such that the 2-core of p is 1, and the 2-quotient of p is (p0, p1), in particular, we have |p| =2n +1=1+2(|p0| + |p1|).

4. EXAMPLE OF CORRESPONDENCE BETWEEN REPRESENTATIONS OF S2n AND S2n+1 In the previous section, we have defined a basechange map from irreducible repre- sentations of Bn to irreducible representations of S2n as well as S2n+1, which induces a bijection between certain irreducible representations of S2n with certain irreducible repre- sentations of S2n+1, more precisely there is a bijection between irreducible representations of S2n which correspond to partitions which have empty 2-core with those irreducible rep- resentations of S2n+1 which correspond to partitions which have 2-core (1), the bijection is defined by demanding the partitions of 2n and 2n +1 have the same 2-quotient. In this section, as an example, we tabulate the bijection of partitions of 8 with empty 2-core to partitions of 9 with core (1), inducing a map on representations of the corre- sponding symmetric groups. We have illustrated this to point out that what is an obvious map from the set of partitions of 2n to set of partitions of 2n +1 from the point of view of 2-core and 2-quotient, has no obvious formulation from the point of view of the partitions ′ directly. The table below also gives the character values Θ(w0) and Θ (w0) for the re- ′ spective representations of S8 and S9, once again, the character values Θ(w0) and Θ (w0) which are nonzero are the same up to a sign, but the sign is not obvious by looking at the table, but for which we did give a recipe earlier. 8 FRANKLUBECK¨ AND DIPENDRA PRASAD

′ Θ(w0)=Θ (w0)=1 [18] −−−−−−−−−−−−−−−→ [19]

′ Θ(w0)=−Θ (w0)=−1 [2, 16] −−−−−−−−−−−−−−−→ [3, 2, 14]

′ Θ(w0)=−Θ (w0)=4 [22, 14] −−−−−−−−−−−−−−−→ [3, 16]

′ Θ(w0)=−Θ (w0)=−4 [23, 12] −−−−−−−−−−−−−−−→ [3, 23]

′ Θ(w0)=−Θ (w0)=6 [24] −−−−−−−−−−−−−−−→ [3, 22, 12]

′ Θ(w0)=−Θ (w0)=−3 [3, 15] −−−−−−−−−−−−−−−→ [22, 15]

′ Θ(w0)=−Θ (w0)=−2 [3, 22, 1] −−−−−−−−−−−−−−−→ [24, 1]

′ Θ(w0)=Θ (w0)=8 [32, 12] −−−−−−−−−−−−−−−→ [32, 13]

′ Θ(w0)=−Θ (w0)=−6 [32, 2] −−−−−−−−−−−−−−−→ [33]

′ Θ(w0)=−Θ (w0)=3 [4, 14] −−−−−−−−−−−−−−−→ [5, 2, 12]

′ Θ(w0)=−Θ (w0)=−6 [4, 2, 12] −−−−−−−−−−−−−−−→ [5, 14]

′ Θ(w0)=Θ (w0)=8 [4, 22] −−−−−−−−−−−−−−−→ [5, 22]

′ Θ(w0)=−Θ (w0)=−2 [4, 3, 1] −−−−−−−−−−−−−−−→ [5, 4]

′ Θ(w0)=−Θ (w0)=6 [42] −−−−−−−−−−−−−−−→ [5, 3, 1]

′ Θ(w0)=−Θ (w0)=3 [5, 13] −−−−−−−−−−−−−−−→ [4, 2, 13]

′ Θ(w0)=−Θ (w0)=−4 [5, 3] −−−−−−−−−−−−−−−→ [42, 1]

′ Θ(w0)=−Θ (w0)=−3 [6, 12] −−−−−−−−−−−−−−−→ [7, 2]

′ Θ(w0)=−Θ (w0)=4 [6, 2] −−−−−−−−−−−−−−−→ [7, 12]

′ Θ(w0)=−Θ (w0)=−1 [7, 1] −−−−−−−−−−−−−−−→ [6, 2, 1]

′ Θ(w0)=Θ (w0)=1 [8] −−−−−−−−−−−−−−−→ [9]

The above table misses out the partitions [5, 2, 1] and [3, 2, 13] of 8, and partitions [8, 1], [6, 3], [6, 13], [4, 3, 2], [4, 3, 12], [4, 22, 1], [32, 2, 1], [2, 17], [23, 13], [4, 15] of 9, where ′ Θ(w0)=Θ (w0)=0. A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP 9

5. A THEOREM OF LITTLEWOOD This section aims to give a proof of a theorem due to Littlewood, see pages 143-146 of [Li1], on character values for symmetric groups at exactly the same set of conjugacy classes that we have considered in this paper: product of disjoint even cycles without fixed points — Littlewood’s theorem covers only the case of S2n, which we prove also for S2n+1. Our main theorem proved in the next section is a simple consequence of the theorem of Littlewood (suitably extended to S2n+1). We have decided to include a proof of the theorem due to Littlewood since the proof (from 1940!) is hard to follow. All the proofs on character values of representations of the symmetric group eventually boil down to a theorem of Frobenius, which is a consequence of Schur-Weyl duality, turning theorems about character theory of GLn(C) to character theory for the symmetric group. We begin by recalling the relationship between the two character theories: it is remarkable that they are the same! Let Rd be the representation ring of the symmetric group Sd, treated here only as an ∞ abelian group. Let R = Rd, together with the multiplication Rn ⊗ Rm → Rn+m Pd=0 which corresponds to induction of a representation (V1 ⊠ V2) of Sn × Sm to Sn+m. These multiplications turn R into a commutative and associative graded ring. It can be seen that ∼ as graded rings, R = Z[H1, ··· ,Hd, ··· ], the polynomial ring in infinitely many variables Hi, i ≥ 1, where each Hi is given weight i, and corresponds to the trivial representation of Si. On the other hand, let Λ = Z[X ,X , ··· ,X ]Sn = Λk , n 1 2 n M n k≥0 k where Λn is the space of symmetric polynomials in Z[X1,X2, ··· ,Xn] of degree k. Define, Λk = lim Λk , ←− n where the inverse limit is taken with respect to natural map of polynomial rings

Sn+1 Sn Z[X1,X2, ··· ,Xn+1] → Z[X1,X2, ··· ,Xn] in which Xn+1 is sent to the zero element. Finally, define the graded ring Λ= Λk. M k≥0 The ring Λ, often by abuse of language (in which we too will indulge in) is called the ring of symmetric polynomials in infinitely many variables, comes equipped with surjective k homomorphisms to Λn for all n ≥ 0, which is in fact an isomorphism restricted to Λ k onto Λn for k ≤ n. As typical elements of the ring Λ, also of paramount importance, note the following symmetric functions of degree m in infinitely many variables X1,X2,X3, ··· : m m (1) pm = X1 + X2 + ··· (an infinite sum),

(2) em = Xi1 Xi2 ··· Xim , where the sum is over all indices i1 < i2 < ··· < im, P (3) hm = Xi1 Xi2 ··· Xim , where the sum is over all indices i1 ≤ i2 ≤···≤ im. P Later, for any partition λ = {λ1 ≥ λ2 ≥···≥ λm} of k, we will have occasion to use the following symmetric functions (in infinite number of variables) of degree k: 10 FRANKLUBECK¨ AND DIPENDRA PRASAD

(1) pλ = pλ1 · pλ2 ··· pλm ,

(2) eλ = eλ1 · eλ2 ··· eλm ,

(3) hλ = hλ1 · hλ2 ··· hλm .

As λ varies over all partitions of k, eλ form a basis of the space of symmetric polyno- mials (in infinitely many variables) of degree k; similarly, hλ form a basis of the space of symmetric polynomials of degree k; whereas pλ forms a basis after Z is replaced by Q as the coefficient ring. ∞ ∞ k Observe that the graded rings R = Rd and Λ = Λ are isomorphic under Pd=0 Pk=0 the map which sends Hm to hm, call this map Ψ. Note that a polynomial representation of GLm(C) has as its character at the diagonal Sm element (X1,X2, ··· ,Xm) in GLm(C), a symmetric polynomial in Z[X1, ··· ,Xm] . A nice fact about irreducible polynomial representations of GLm(C), say with highest weight λ1 ≥ λ2 ≥···≥ λm is that it makes sense to speak of “corresponding irreducible representations” of GLd(C) for all d ≥ m by extending the highest weight λ1 ≥ λ2 ≥ ··· ≥ λm by adding a few zeros after λm. The characters of these representations of Sd GLd(C) for d ≥ m, which are symmetric polynomials in Z[X1,X2, ··· Xd] , correspond Sd Sd′ ′ to each other under the maps: Z[X1,X2, ··· Xd] → Z[X1,X2, ··· Xd′ ] for d ≥ d ≥ m. Therefore, a polynomial representation of GLm(C) has as its character an element ∞ k Sm which can be considered to belong to Λ= Λ (and not only in Z[X1, ··· ,Xm] ). Pk=0 For example, the character of the standard m-dimensional representation of GLm(C) is the infinite sum X1 + X2 + ··· . So far, nothing non-obvious has been said. Now, here is a non-obvious fact, a form of the Schur-Weyl duality, that if λ = {λ1 ≥ λ2 ≥···≥ λm} is a partition of n, defining an irreducible representation of Sn, say πλ, and defining at the same time an irreducible representation of GLd(C) with highest weight {λ1 ≥ λ2 ≥···≥ λm ≥ 0 ≥···≥ 0} for all d ≥ m, with its character, the Schur function Sλ, then

Ψ(πλ)= Sλ.

(It is helpful to recall that under the Schur-Weyl duality, the trivial representation of Sm m d goes to the irreducible representation Sym (C ) of GLd(C) corresponding to the parti- tion {1, 1, ··· , 1} of m.) A form of the Frobenius theorem on characters of the symmetric group will play an im- portant role in this work, so we recall this as well as supply a proof in the next Proposition. (See for example [F-H], Exercise 4.52(e), for the statement of this result.)

Proposition 5.1. For λ = {λ1 ≥ λ2 ≥ ··· ≥ λm}, a partition of n, defining an irre- ducible representation πλ of Sn, and defining at the same time an irreducible representa- tion of GLd(C) with highest weight {λ1 ≥ λ2 ≥···≥ λm ≥ 0 ≥···≥ 0} for all d ≥ m, with its character, the Schur function Sλ, we have the following identity: Θ (c ) (1) S = λ ρ p , λ X |Z(c )| ρ ρ ρ where ρ = (ρ1, ρ2, ··· ) is a partition of n, defining a conjugacy class cρ in Sn, whose centralizer in Sn is Z(cρ) of order |Z(cρ)|; the symmetric function pρ is the product of ρi ρi ρi symmetric polynomials pρi = X1 + X2 + X3 + ··· . A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP11

Proof. We will prove a generalised version of the identity being proved: Θ (c ) (1) S = λ ρ p , λ X |Z(c )| ρ ρ ρ to the following identity among symmetric functions: Θ (c ) (2) Ψ(π)= π ρ p , X |Z(c )| ρ ρ ρ where π is any representation of a symmetric group Sn, and Ψ(π) is the associated sym- metric function of degree n in infinitely many variables arising through the isomorphism Ψ: R → Λ. (The symmetric function Ψ(π) can of course be identified to the character of a representation of GLm(C).) Once (2) is proved, appealing to the Schur-Weyl duality according to which the isomorphism of graded rings Ψ: R → Λ which is defined by sending the trivial representation Hn of Sn to the symmetric polynomial hn (in infinitely many variables) takes the representation πλ to the Schur function Sλ proves (1) and hence the Proposition. Since the identity (2) is linear in the representation π, it suffices to check it on a set of linear generators (over Z, although it suffices to do it for generators over Q too) of the ring R. Our proof of identity (2) will therefore be accomplished in two steps.

(1) Prove that the identity (2) holds for the trivial representation Hn of Sn. (2) If the identity (2) holds for a representation π1 of Sm and π2 of Sn, then it also holds good for the representation π1 ×π2 of Sm+n induced from the representation π1 ⊗ π2 of Sm × Sn ⊂ Sm+n. We begin by proving step 1, i.e. that the identity (2) holds for the trivial representation Hn of Sn. m For any integer m, let Vm = C be the m-dimensional vector space over C. By k definition, hk is the character of the representation Sym (Vm) assuming that m ≥ k; in fact the integer m will play no role as long as it is large enough, preferably infinity! Thus, ∞ 1 tnh = . X n (1 − tX )(1 − tX ) ··· (1 − tX ) n=0 1 2 m Taking logarithm on the two sides, ∞ m log( tnh ) = − log(1 − tX ), X n X i n=0 i=1 t2 t3 = t( X )+ ( X2)+ ( X3)+ ··· X i 2 X i 3 X i Now taking exponential of the two sides: ∞ t2 2 t3 3 tnh = expt(P Xi) · exp 2 (P Xi ) · exp 3 (P Xi ) ··· X n n=0 (tp )2 (t2p /2)2 (t3p /3)2 = [1 + tp + 1 + ··· ][1 + t2p /2+ 2 + ··· ][1 + t3p /3+ 3 + ··· ] 1 2! 2 2! 3 2! Therefore, 12 FRANKLUBECK¨ AND DIPENDRA PRASAD

i1 i2 i3 p1 (p2/2) (p3/3) hn = ··· X i1! i2! i3! pi1 · pi2 · pi3 ··· =∗ 1 2 3 i2 i3 X i1!2 i2!3 i3! ··· where the summation is taken over i1 +2i2 + ··· = n. This proves the identity (2) for the trivial representation Hn of Sn for which Θπ(cρ) is identically 1, and the denominators in the right hand side of the equality (∗) are the order of the centralizer of the conjugacy class cρ ∈ Sn. Next, we prove that if the identity (2) holds for a representation π1 of Sm and π2 of Sn, then it also holds good for the representation π1 × π2 of Sm+n induced from the representation π1 ⊗ π2 of Sm × Sn. For this, we slightly rewrite the identity (2) as: 1 (3) Ψ(π)= Θ (x)p , n! X π x x∈Sn where π is a representation of the symmetric group Sn with Θπ(x) its character at an element x ∈ Sn, and px is what was earlier denoted as pρ(x) where ρ(x) denotes the partition of n associated to x. Because Ψ(π1 × π2)=Ψ(π1) · Ψ(π2), assuming that (3) holds for π1 and π2, to prove that it also holds for π1 × π2, we are reduced to proving:

1 1 (4) Θ (g)p = Θ (h )Θ (h )p p . (n + m)! X π1×π2 g m!n! X π1 1 π2 2 h1 h2 g∈Sm+n (h1,h2)∈Sm×Sn

This will be a simple consequence of the character of the induced representation π1 × π2 Sm+n of Sm+n given by: π1 × π2 = IndSm×Sn (π1 ⊗ π2). Note the well-known identity regarding the character f ′ of the induced representation G IndH (U) of a representation U of H with character f (a function on H, extended to a function on G by declaring it zero outside H) at an element s ∈ G: 1 f ′(s)= f(t−1st). |H| X t∈G It follows that for any conjugacy class function λ(s) on G, 1 1 (5) f ′(s)λ(s)= f(t)λ(t). |G| X |H| X s∈G t∈H

Sm+n Now (3) follows from applying (5) to the induced representation π1 ×π2 = IndSm×Sn (π1 ⊗ π2), and where λ(s) = ps for s ∈ Sm+n are symmetric functions used before (products i1 i1 i1 of X1 + X2 + X3 + ··· ). (We are thus applying the identity (5) for λ(s) not a scalar valued function on G, but rather with values in the ring of symmetric polynomials, which we leave to the reader to think about!) A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP13

We have thus completed the proof of the Proposition. 

Before proceeding further, we need to introduce a sign ǫ(p) associated to any partition with either empty 2-core or with 2-core 1. Definition 5.2. (Sign of a partition) (a) For a partition p of an even number with even number of parts 2m (by adding a zero at the end if necessary) with β-numbers β(p) = 0 {β2m−1 > β2m−2 > ··· > β1 > β0}, and with the even and odd β-numbers: β (p) and 1 β (p) of the same cardinality. Let X2m = {2m − 1, 2m − 2, ··· , 1, 0}, and let i(p) be the bijection from X2m to β(p) sending i → βi. Let j(p) be the unique order preserving bijective map from the ordered set β0(p) ∪ β1(p) (in which there is no order relationship between odd and even numbers) to the ordered set X2m taking even numbers to even numbers, and odd numbers to odd numbers. The permutation s(p) on X2m defined as the i(p) j(p) composition of the maps X2m → β(p) → X2m will be called the shuffle permutation associated to p, and its sign (−1)s(p) will be denoted ǫ(p). (b) For a partition p of an odd number with odd number of parts 2m +1 (by adding a zero at the end if necessary) with β-numbers β(p) = {β2m > β2m−1 > ··· > β1 > 0 1 β0}, and with the even and odd β-numbers: β (p) and β (p) with the number of odd β- numbers m+1, let X2m+1 = {2m+1, 2m, 2m−1, ··· , 1} (let’s be reminded that X2m+1 is different from X2m, and in particular, it has more odd numbers than even numbers). Let i(p) be the bijection from X2m+1 to β(p) sending i → βi−1. Let j(p) be the unique order preserving bijective map from the ordered set β0(p) ∪ β1(p) (in which there is no order relationship between odd and even numbers) to the ordered set X2m+1 taking even numbers to even numbers, and odd numbers to odd numbers. The permutation s(p) on i(p) j(p) X2m+1 defined as the composition of the maps X2m+1 → β(p) → X2m will be called the shuffle permutation associated to p. Define, ǫ(p)=(−1)m(−1)s(p). Remark 5.3. The sign defined here, see Proposition 5.4 below for a much simpler defi- nition, will eventully be the sign of the associated respresentation of the symmetric group S|p| at the element w0 = (12)(34) ··· (2n−1, 2n) inside S2n or S2n+1 for |p| =2n, 2n+1. The proof of the following proposition will be given by Arvind Ayyer in Appendix . Proposition 5.4. For a partition p of an integer n, let 2k or 2k + 1, respectively, be the number of odd entries in p. Associated to the partition p, we define another sign ′ k ǫ (p) := (−1) . If the partition p corresponds to an irreducible representation π of S|p|, then we define ǫ′(π)= ǫ′(p). The the following holds:

ǫ(p)= ǫ′(p). Next, we recall the following theorem from [Li1] where it is contained in equations 7.3.1 and 7.3.2 of page 132; it is also contained in [P] as Theorem 2. Both [Li1] as well as [P] deal with a more general theorem involving GLmn(C). This theorem as well as the next theorem should be considered as the GLn(C) analogues of theorems on character values that we strive to prove here. It is in this theorem that the notion of p-core and p-quotients make their appearance (stated here only for p = 2, although stated more generally in [P]). 14 FRANKLUBECK¨ AND DIPENDRA PRASAD

Theorem 5.5. Let λ = {λ1 ≥ λ2 ≥···≥ λ2m} be the highest weight of an irreducible polynomial representation Vλ of GL2m(C) with character Sλ. Let

X =(x1, x2, ··· , xm, −x1, −x2, ··· , −xm)=(X, −X), × be a diagonal matrix in GL2m(C) with xi ∈ C arbitrary. Then if Sλ(X) is not identically 0 1 zero, its 2-core c2(λ) must be empty, in which case, if {λ ,λ } is the 2-quotient of λ, then, 2 2 Sλ(X)= ǫ(λ)Sλ0 (X )Sλ1 (X ), where Sλ0 and Sλ1 are the characters of the corresponding highest weight modules of 2 GLm(C), X is the diagonal matrix (x1, x2, ··· , xm), and X its square. Proof. For the sake of completeness, we give the proof. Write the matrix whose determi- nant represents the numerator of the Weyl character formula as (where βi = λi +2m − i):

β1 β1 β1 β1 β1 x1 x2 ··· xm (−x1) ··· (−xm)  β2 β2 β2 β2 β2  x1 x2 ··· xm (−x1) ··· (−xm)  ···········  ·    ···········   β2m β2m β2m β2m β2m   x1 x2 ··· xm (−x1) ··· (−xm) 

In this 2m×2m-matrix, adding the first m columns of the matrix to the last m columns, we find that all the rows of the last m columns with βi odd become zero, and those rows with βi even get multipliedby 2. In the new matrix, substracting the half of last m columns to the first m columns, makes all rows in the first m columns with βi even to be zero. Let d be the number of βi which are odd, and therefore 2m − d is the number of βi which are even. Thus we get a matrix in which in the first m columns, there are exactly d nonzero rows, and in the last m columns, there are exactly 2m − d complementary rows which are nonzero. The of such a matrix is nonzero only for d = 2m − d, i.e., d = m, and in which case, by permuting rows, we come to a block diagonal matrix which looks like (where γk, δℓ are the odd and even numbers among βi written in decreasing order):

γ1 γ1 γ1 x1 x2 ··· xm 0 0 ··· 0  γ2 γ2 γ2  x1 x2 ··· xm 0 0 ··· 0  ············   γm γm γm   x1 x2 ··· xm 0 0 ··· 0   δ1 δ1 δ1  ·  0 0 ··· 0 x1 x2 ··· xm   δ2 δ2   0 0 ··· 0 x x ··· xδ2   1 2 m   ············   δm δm δm   0 0 ··· 0 x1 x2 ··· xm 

The determinant of this matrix is the product of two Weyl numerators for GLm(C), δi 2 δi/2 γi in which δi being even, we write xj as (xj ) , and γi being odd, we write xj as 2 (γi−1)/2 (xj ) · xj. We have a similar factorization of the Weyl denominator. Next we observe that the deteriminant of a matrix in which the rows have been shuffled using a permutation σ of the rows, changes by multiplication by the sign (−1)σ. This A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP15 needs to be done both for the numerator which involves the ‘shuffle permutation’ intro- duced earlier in Definition 5.2, and the denominator for which the shuffle permutation is identity matrix; to be more precise, the numerator needs the permutation s(p)s0, and the denominator needs s0 where s0 is the permutation of X2m = {2m − 1, 2m − 2, ··· , 1, 0} sending odd numbers in X2m consecutively to {2m−1, 2m−2, ··· , m} and even numbers in X2m consecutively to {m − 1, ··· , 1, 0}. This completes the proof of the theorem.  We will also need the following variant of the previous theorem which is proved as the previous theorem by manipulations with the explicit character formula for GLn(C) as quotients of two ; we will not give a proof of this theorem.

Theorem 5.6. Let λ = {λ1 ≥ λ2 ≥···≥ λ2m+1} be the highest weight of an irreducible polynomial representation Vλ of GL2m+1(C) with character Sλ. Let

X =(x1, x2, ··· , xm, −x1, −x2, ··· , −xm, x)=(X, −X, x), × be a diagonal matrix in GL2m+1(C) with xi, x ∈ C arbitrary. Then if Sλ(X) is not identically zero, then the number of even entries and number of odd entries in the β- sequence β(λ)= {β1 = λ1 +2m > β2 = λ2 +2m − 1 > ··· >λ2m+1}, differ by one. If the number of odd entries in the β-sequence is one more than the number of even entries, then its 2-core is {1}, whereas if the number of even entries in the β-sequence is one more than the number of odd entries, then its 2-core is empty. In either case, let {λ0,λ1} be the 2-quotient of λ. Then,

2 2 2 (a) Sλ(X)= ǫ(λ) · x · Sλ0 (X )Sλ1 (X , x ) ifthe number of odd entriesin the β-sequence is one more than the number of even entries; whereas 2 2 2 (b) Sλ(X)= ǫ(λ)Sλ1 (X )Sλ0 (X , x ) if the number of even entries in the β-sequence is one more than the number of odd entries. Here Sλ0 and Sλ1 are the characters of the corresponding highest weight modules of GLm(C), and GLm+1(C) in case (a), and of GLm+1(C), and GLm(C) in case (b). The following theorem is due to Littlewood [Li1], pages 143-146, for even symmetric groups. Forms of this theorem seem to be available in the literature more generally for n arbitrary (Z/p) ⋊ Sn contained in Spn, such as in Theorem 4.56 of [Ro] which also follows from Theorem 2 of [P] and the corresponding analogue of Theorem 5.6 for general p. Theorem 5.7. Let p → π(p) be the natural correspondence between partitions/Young diagram and irreducible representations of the symmetric group S|p|. Then if the repre- sentation π(p) is to have nonzero character at some element w in S|p| which is a product of disjoint even cycles together with at most one fixed point, the 2-core of p must be empty if |p| is even, and 2-core must be 1 if |p| is odd. Further, if w = w1 ··· wd, product of disjoint cycles wi of lengths 2ℓi together with a cycle of length ≤ 1, then, ′ Θ(π(p))(w)= ǫ(p)Θ(π(p0) × π(p1))(w ), 16 FRANKLUBECK¨ AND DIPENDRA PRASAD

′ ′ ′ where w = w1 ··· wd is an element in Sn, n = [|p|/2] which is a product of disjoint cycles of length ℓi, and where we have used the notation π(p0) × π(p1) for the representation (usually reducible) of Sn Sn π(p0) × π(p1) = Ind (π(p0) ⊠ π(p1)). S|p0|×S|p1|

Proof. Recall once again the notation that if λ = {λ1 ≥ λ2 ≥···≥ λm} is a partition of k, it defines an irreducible representation of Sk, say πλ with character Θλ, and defines at the same time an irreducible representation of GLd(C) for all d ≥ m of highest weight {λ1 ≥ λ2 ≥···≥ λm ≥ 0 ≥···≥ 0}, with its character, the Schur function Sλ, and under the isomorphism Ψ: R → Λ,

Ψ(πλ)= Sλ. By Proposition 5.1, Θ (c ) (1) S = λ ρ p , λ X |Z(c )| ρ ρ ρ where ρ = (ρ1, ρ2, ··· ) is a partition of k, defining a conjugacy class cρ in Sk, whose centralizer in Sk is Z(cρ) of order |Z(cρ)|; the symmetric function pρ is the product of ρi ρi ρi symmetric polynomials pρi = X1 + X2 + X3 + ··· i1 i2 ik In the standard notation, if a conjugacy class c in Sk is (1 , 2 , ··· ,k ), the order of the centralizer of any element in c is:

i2 ik (i1)!2 (i2)! ··· k (ik)!.

For a partition ρ =(ρ1, ρ2, ··· ) of k, we will be using the notation 2ρ for the partition 2ρ = (2ρ1, 2ρ2, ··· ) of 2k, defining a conjugacy class c2ρ in S2k for which the order of the centralizer of any element in c2ρ is:

i1 i2 ik 2 (i1)!4 (i2)! ··· (2k) (ik)!, therefore, p (2) |Z(c2ρ)| =2 |Z(cρ)|, where p = i1 + i2 + ··· + ik. Now we split the proof of the theorem in two cases. Case 1: |p| =2n. We will use the identity expressed by equation (1) at the diagonal matrices in GL2m(C) of the form X = (x1, x2, ··· , xm, −x1, −x2, ··· , −xm)=(X, −X). An important ob- ρi ρi servation is that pρ which is the product of the symmetric polynomials pρi =(X1 +X2 + ρi X2 + ··· ) must be identically zero on such elements unless all the entries in ρ are even. Using Theorem 5.5, we write equation (1) as:

2 2 Θλ(c2ρ) ǫ(λ)S 0 (X )S 1 (X )= p (X). λ λ X |Z(c )| 2ρ ρ 2ρ p 2 Since p2ρ(X)=2 pρ(X ), where p = i1 + i2 + ··· + in, we can rewrite this equation using (2) as:

Θλ(c2ρ) (3) ǫ(λ)S 0 (X)S 1 (X)= p (X). λ λ X |Z(c )| ρ ρ ρ A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP17

Since Ψ: R → Λ is an isomorphism of rings, the element Sλ0 (X)Sλ1 (X) of Λ arises from the image under Ψ of the representation of Sn that we have called π(p0) × π(p1) in the statement of Theorem 5.7, therefore, we have, Θ (c ) Θ (c ) (4) ǫ(λ) π(p0)×π(p1) ρ p (X)= λ 2ρ p (X). X |Z(c )| ρ X |Z(c )| ρ ρ ρ ρ ρ (By linearity, the form of Frobenius theorem on characters in Proposition 5.1 in equation (1) is valid for all representations – not necesarily irreducible – of symmetric groups.) Since the polynomials pρ(X) are linearly independent, we can equate the coefficients of pρ(X) on the two sides of equation (4) to prove the theorem when |p| is even. By Theorem 5.5, if λ has non-empty 2-core, then Sλ is identically zero on the set of diagonal elements of the form X = (x1, x2, ··· , xm, −x1, −x2, ··· , −xm)=(X, −X). By the linear independence of the polynomials pρ(X), we deduce that Θλ(c2ρ) ≡ 0.

Case 2: |p| =2n +1. In this case, we will use Frobenius character relationship contained in Proposition 5.1: Θ (c ) (5) S = λ ρ p , λ X |Z(c )| ρ ρ ρ at diagonal elements of GL2m+1(C) of the form:

X =(x1, x2, ··· , xm, −x1, −x2, ··· , −xm, x)=(X, −X, x), × with xi, x ∈ C arbitrary. Observe that if ℓ is an odd integer, then for X as above, ℓ ℓ pℓ(X)= x , whereas for ℓ an even integer pℓ(X)=2pℓ(X)+ x . It follows that for any conjugacy class ρ in S2n+1, pρ, and hence each term in the right hand side of equation (5) is divisible by x, and all terms except those which correspond to those ρ which are product of disjoint even cycles together with exactly one fixed point, contribute a term which is divisible by x and no higher power, and all the other terms are divisible by higher powers of x. Furthermore, in the case |p| =2n +1, each of the term pρ(X)/x is an even function of x, hence by equation (5), Sλ(X) as a function of x is an odd polynomial function of x. Thus, if Sλ(X) is nonzero, we must be in case (a) of Theorem 5.6 (since in case (b) Sλ(X) is an even function of x), therefore if Sλ(X) is a nonzero function of x, λ must have 2-core {1}. By Theorem 5.6, the left hand side of the equation (5) is also divisible by x which after 2 2 2 dividing by x gives, ǫ(λ)Sλ0 (X )Sλ1 (X , x ). Thus after dividing both the sides of the Frobenius character relationship in equation (5) by x, and then putting x = 0, we are in exactly the same situation as in the proof of the theorem for S2n, for which we do not need to repeat the previous argument. 

6. THE THEOREM The following theorem is the main result of this paper and is a simple consequence of Theorem 5.7 of the last section.

Theorem 6.1. Let G be either the symmetric group S2n or the symmetric group S2n+1, n and H the hyperoctahedral group Bn = (Z/2) ⋊ Sn, sitting naturally inside G as the 18 FRANKLUBECK¨ AND DIPENDRA PRASAD centralizer of the involution w0 = (1, −1)(2, −2) ··· (n, −n). Then there exists an injec- tive map BC from the set of irreducible representations of Bn to the set of irreducible representations of G, and a map Nm : X → Bn defined and discussed in §2 on a subset X ⊂ G consisting of those elements in G which are disjoint product of even cycles and with at most one fixed point such that, (using the notation Θ(V )(g) for the character of a representation V of a group G at an element g ∈ G) Θ(BC(π))(w)= ǫ(BC(π))Θ(π)(Nmw), for any irreducible representation π of Bn, and w ∈ X ⊂ G. In particular, taking w = w0, we have Θ(BC(π))(w0)= ǫ(BC(π))Θ(π)(1).

Thus the character of an irreducible representation of G at the element w0 is, up to a n sign, the dimension of an irreducible representation of Bn = (Z/2) ⋊ Sn. Further, the image of the map π → BC(π) from the set of irreducible representations of Bn to the set of irreducible representations of G consist exactly of those representations of G whose character at w0 is nonzero. Proof. The proof of this theorem is a simple consequence of Theorem 5.7 of the last section. Recall that we have used the notation π(p0) × π(p1) for the representation Sn π(p0) × π(p1) = Ind (π(p0) ⊠ π(p1)). S|p0|×S|p1|

Of course the representation π(p0) × π(p1) of Sn is a complicated sum of irreducible representations for which there is the Littlewood-Richardson rule. However, this compli- cation has no role to play for us since instead of Sn we are dealing with the larger group, n Bn =(Z/2) ⋊ Sn and the representation π(p0) × π(p1) of Sn is the restriction to Sn of a representation of Bn that basechanges to the representation π of S2n. n Observe that the representation of Bn =(Z/2) ⋊ Sn that we are dealing with is Bn IndA (V ), n where A is the subgroup of Bn which is A = (Z/2) ⋊ (S|p0| × S|p1|), and V is the representation of A on which (Z/2)n =(Z/2)|p0|+|p1| =(Z/2)|p0| × (Z/2)|p1| acts by the trivial character on the first factor (Z/2)|p0|, and by the non-trivial character |p1| ⊠ on each of the Z/2 factors in (Z/2) ; the subgroup S|p0| × S|p1| acts by π(p0) π(p1). By standard application of Mackey theory (restriction to Sn of an induced representa- tion of Bn), we find that Bn Sn ⊠ ResSn (Ind (V )) = Ind (π(p0) π(p1)) = π(p0) × π(p1), A S|p0|×S|p1| proving the theorem.  Here are a few samples that we first did using GAP software [GAP].

Example 1: The symmetric group S20 has 627 irreducible representations, and the char- acter values Θ(w0) is: (1) > 0 for 227 representations, (2) < 0 for 254 representations, A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP19

(3) 0 for 146 representations.

On the other hand, B10 has 227+254 = 481 irreducible representations, of dimensions exactly |Θ(w0)|.

Example 2: The symmetric group S21 has 792 irreducible representations, and the char- acter values Θ(w0) is: (1) > 0 for 252 representations, (2) < 0 for 229 representations, (3) 0 for 311 representations.

On the other hand, B10 has 252+229 = 481 irreducible representations, of dimensions exactly |Θ(w0)|.

Acknowledgement: The authors must thank ICTS-TIFR, Bengaluru for the invitation to the program “Group Algebras, Representations and Computation” in October 2019 which brought the two authors together to collaborate on this project. The second author thanks Dr Ashish Mishra for his tutorials on p-core and p-quotients, and in particular, for making the paper [Li2] available to him. The authors thank Arvind Ayyer for supplying the appendix proving Proposition 5.4.

REFERENCES [D] F.Digne: Shintani descent and L-functions on Deligne Lusztig varieties, Proc. Sympos. Pure Math. 47, part 1, (1987), 61-68. [F-H] W. Fulton and J. Harris: Representation theory, a first course, Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. [GAP] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.2; 2019, https://www.gap-system.org. [Ka] N. Kawanaka, Liftings of irreducible characters of finite classical groups. I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 851-861 (1982). [KLP] S. Kumar,G. Lusztig andD. Prasad, Characters of simplylaced nonconnected groups versus char- acters of nonsimplylaced connected groups. Contemporary Math., vol 478, AMS, pp. 99-101. [Li1] D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups. Oxford University Press, New York 1940. [Li2] D. E. Littlewood, Modular representations of symmetric groups. Proc. Roy. Soc. London Ser. A 209 (1951), 333-353. [Lu] G.Lusztig. Left cells in Weyl groups. Lie group representations, I (College Park, Md., 1982/1983), 99-111, Lecture Notes in Math., 1024, Springer, Berlin, 1983. [P] D.Prasad, A character relationship on GLn(C); Israel Journal, vol. 211, (2016), 257-270. [Ro] G.deB.Robinson, Representation theory of the symmetric group. Mathematical Expositions, No. 12. University of Toronto Press, Toronto 1961 vii+204 pp. [Sh] T. Shintani: Two remarks on irreducible characters of finite general linear groups. J. Math. Soc. Japan 28 (1976), no. 2, 396-414. 20 FRANKLUBECK¨ AND DIPENDRA PRASAD

APPENDIX A. PROOF OF PROPOSITION 5.4 BY ARVIND AYYER

Let p =(p1,..., pn) be a partition of n, written as usual with p1 ≥···≥ pn. We will think of p as a partition with n parts by padding with zeros on the right if necessary. For example, the set of partitions of 3 is {(3, 0, 0), (2, 1, 0), (1, 1, 1)}. Then, the β-set of p, is given by β(p)=(β1,...,βn), where βi = pi +n−i. Then we reformulate Proposition 3.1 as follows.

Proposition A.1. (1) The partition p = (p1,..., p2n) of 2n has empty 2-core if and only if there are exactly n odd integers in β(p). (2) The partition p =(p1,..., p2n−1) of 2n − 1 has 2-core equal to (1) if and only if there are exactly n odd integers in β(p). Recall that, given a partition p, the shuffle permutation in Definition 5.2 is the element π ∈ Sn mapping odd (resp. even) elements of β(p) to odd (resp. even) elements of (n−1,..., 1, 0) preserving their order. (Observe that any totally ordered set of n elements is canonically isomorphic to the set {1, 2, ··· , n}, and therefore a bijection between two totally ordered sets of order n gives rise to an element of Sn.) The sign of the shuffle permutation is denoted ǫ(p) = sgn(π). It is easy to see that if p is a partition of 2n (resp. 2n − 1), then p has an even (resp. odd) number of odd parts. The restatement of Proposition 5.4 is then the following. Proposition A.2. (1) For p ⊢ 2n with empty 2-core, let 2k be the number of odd parts in p. Then ǫ(p)=(−1)k. (2) For p ⊢ 2n − 1 with 2-core equal to (1), let 2k − 1 be the number of odd parts in p. Then ǫ(p)=(−1)k+n−1.

Proof. We begin with the proof of case (1). By Proposition A.1, exactly n of these βi’s are odd. Let the integers i such that βi is odd be labeled o1 < ···

v1 ∧ v2 ∧···∧ vn ∧ vn+1 ∧···∧ v2n = ǫ1vo1 ∧ vo2 ∧···∧ von ∧ ve1 ∧···∧ ven , which means that ǫ1 has the parity of:

(n − e1 +1)+(n − e2 +2)+ ··· +(n − en + n)=(o1 + ··· + on) − (1 + ··· + n).

By the computation just done, if ǫ2 is the sign of the permutation that moves all the odd integers 1 ≤ i ≤ (2n − 1) to positions 1,...,n (preserving their order), and even integers 2 ≤ i ≤ 2n to positions n +1,..., 2n (preserving their order), then ǫ2 is given by:

=(o1 + ··· + on) − (1 + ··· + n)=(1+3+ ··· +2n − 1) − (1 + ··· + n).

Therefore, we find that ǫ(p)= ǫ1ǫ2 has the same parity as (o1 + ··· + on)+ n. Thus, we have to prove that the parity of o1 + ··· + on + n is the same as that of k. Now, since

βoi = poi +2n − oi is odd, so is poi − oi. Thus, the parity of oi +1 is the same as that of poi . As a result, we have to show that the parity of po1 + ··· + pon is the same as that A CHARACTER RELATIONSHIP BETWEEN SYMMETRIC GROUP AND HYPEROCTAHEDRAL GROUP21 of k. Among all poi ’s, those which are even clearly do not contribute to the parity, and those which are odd contribute a 1. We thus have to prove that the cardinality of the set

S = {i | poi is odd} has the same parity as that of k. We will prove something stronger, namely that |S| = k. Suppose that |S| = j. Partition the set {1,..., 2n} = O ∪ E, where O = {o1,...,on}. By assumption, S ⊂ O are the positions where p takes odd values. For convenience, let δi = 2n − i for 1 ≤ i ≤ 2n so that βi = pi + δi. There are exactly n even and n odd δ’s. Since βi for i ∈ O is odd, δi’s for i ∈ S are even and δi’s for i ∈ O \ S are odd. Which means that there are j even and n − j odd δi’s for i ∈ O. Consequently, we must have n − j even and j odd δi’s for i ∈ E. Combining all this information, we infer that the number of odd parts in p is j (from S) and j (from E, since βi’s for i ∈ E are even). But we had assumed that p has 2k odd parts, and therefore j = k, proving first case of the Proposition. Now we consider the second case of the Proposition where the length of the partition p is 2n − 1. In this case, arguing just as in case (1), we have to prove that the parity of o1 + ··· + on is the same as that of k +1.

Now, since βoi = poi +2n − 1 − oi is odd, so is poi − oi − 1. Thus, the parity of oi is the same as that of poi . As a result, we have to show that the parity of po1 + ··· + pon is the same as that of k +1. Among all poi ’s, those which are even clearly do not contribute to the parity, and those which are odd contribute a 1. We thus have to prove that the cardinality of the set S = {i | poi is odd} has the same parity as that of k +1. We will again prove something stronger, namely that |S| = k+1. Suppose that |S| = j. Partition the set {1,..., 2n − 1} = O ∪ E, where O = {o1,...,on}. By assumption, S ⊂ O are the positions where p takes odd values. For convenience, let δi = 2n − 1 − i for 1 ≤ i ≤ 2n − 1 so that βi = pi + δi. There are n even and n − 1 odd δi’s. Since βi for i ∈ O is odd, δi’s for i ∈ S are even and δi’s for i ∈ O \ S are odd. Which means that there are j even and n − j odd δi’s for i ∈ O. Consequently, we must have n − j even and j − 1 odd δi’s for i ∈ E. Combining all this information, we infer that the number of odd parts in p is j (from S) and j − 1 (from E, since βi’s for i ∈ E are even), giving a total of 2j − 1. But we had assumed that p has 2k +1 odd parts, and therefore j = k +1. This completes the proof of both the cases of the Proposition. 

LEHRSTUHL D FUR¨ MATHEMATIK, PONTDRIESCH 14/16, 52062 AACHEN, GERMANY E-mail address: [email protected]

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY, POWAI,MUMBAI-400076

TATA INSTITUTE OF FUNDAMENTAL RESEARCH, COLABA,MUMBAI-400005. E-mail address: [email protected]

INDIAN INSTITUTE OF SCIENCE, BENGALURU-560012 E-mail address: [email protected]