Complex Reflection Groups, Their Irreducible Representations, and a Generalized Robinson-Schensted Algorithm

Complex reﬂection groups, their irreducible representations, and a generalized Robinson-Schensted algorithm Aba Mbirika Department of Mathematics University of Wisconsin–Eau Claire Eau Claire, WI 54701 email: [email protected]

The purpose of this brief overview is to introduce readers to the objects of my research involving complex reflection groups. This introductory material is not intended to provide particular results of my current research. However, it is intended to do the following: (1) Give a brief introduction to complex reflection groups and a variety of ways to view their elements combinatorially. (2) Define the irreducible representations of this infinite family of groups. (3) Explain a generalized Robinson-Schensted algorithm by first looking at an example in the smallest non-trivial case—namely, the hyperoctahedral group. Understanding this simplest setting helps to understand the arbitrary setting introduced next. (4) Give a generalized Robinson-Schensted algorithm for arbitrary complex reflection groups, and provide some related combinatorial implications of this map.

1. Brief introduction to the complex reflection groups G(r, p, n) The family of imprimitive complex reflections groups are defined as follows. Let r, p, n ∈ √ N 2π −1 with p|r and let ζ = exp r . The groups G(r, p, n) are the subgroups of GLn(C) consisting of matrices such that • the entries are either 0 or powers of ζ, • there is exactly one nonzero entry in each row and column, and • the (r/p)-th power of the product of all nonzero entries is 1. Together with 34 exceptional groups, the groups G(r, p, n) account for all finite groups gen- erated by complex reflections [4]. Alternately, we may use the wreath product definition to view G(r, 1, n) as the group of colored permutations in the following manner. The group G(r, 1, n) is defined to be the wreath product of the cyclic group Zr by the symmetric group Sn wherein Zr o Sn := {((a1, . . . , an), σ) | 0 ≤ ai ≤ r − 1, σ ∈ Sn}. The value ai is con- sidered to be the “color” of the entry σi. For an element w ∈ G(r, 1, n), we may use a a1 a2 an r window notation and write this element w as [ζ σ1, ζ σ2, . . . , ζ σn] where ζ = 1 and σ = σ1 ··· σn ∈ Sn. In this notation, w is the matrix whose nonzero entry in the i-th column ai is ζ and appears in row σi. Since G(r, p, n) can be identified as a subgroup of index p of the wreath product, we can use the window notation to denote w ∈ (r, p, n) noting that r G the previous definition forces the product (ζa1 ··· ζan ) p = 1 to hold.

Example 1.1. Let ζ be a primitive 6th root of unity. Consider w = [ζ33, ζ21, ζ32] in 3 2 3 6 G(6, 1, 3). It also lies in G(6, 2, 3) since (ζ ζ ζ ) 2 = 1. However, it does not lie in G(6, 3, 3) 1 3 2 3 6 4 since (ζ ζ ζ ) 3 = ζ 6= 1. In matrix form,  0 ζ2 0  w =  0 0 ζ3  . ζ3 0 0

Observe that the complex reﬂection groups G(r, p, n) give rise to the real reﬂection groups as follows: G(1, 1, n) = W (An−1) (symmetric groups), G(2, 1, n) = W (Bn) (hyperoctahedral groups), G(2, 2, n) = W (Dn) (even signed-permutation groups), and G(r, r, 2) = D2r = W (I2(r)) (dihedral groups).

2. One-dimensional irreducible representations of G(r, p, n) We deﬁne the 2r one-dimensional irreducible representations of G(r, 1, n) following the work of Geck-Jacon [2, Chapter 5]. Let S = {s0, s1, . . . , sn−1} be a generating set given by

so = [ζ · 1, 2, 3, . . . , n] and si = [1, 2, . . . , i − 1, i + 1, i, i + 2, . . . , n], for i ∈ {1, . . . , n − 1}. Then the group G(r, 1, n) admits the following presentation r 2 2 4 3 si | s0, si , (sjsk) , (s0s1) , (slsl+1) , for i ≥ 1, |j − k| > 1, and l ∈ {1, . . . , n − 2} . The subgroups G(r, p, n) are only slightly more complicated and have generating sets in p terms of S consisting of so, s0s1s0, and si for i ∈ {1, . . . , n − 1} (see [1, Section 3]). The 2r irreducible representations of G(r, 1, n) divide naturally into two families.

Deﬁnition 2.1 (Geck-Jacon [2]). For each integer 1 ≤ i ≤ r, deﬁne the representations σi and sgni of G(r, 1, n) by specifying their values on the generating set S as follows: ( ζi−1 if j = 0, and τ (s ) = . i j (−1) if j = 1, . . . , n − 1

0 1 Define σi = τi and sgni = τi . Each becomes a representation of the subgroups G(r, p, n) by restriction. The representations sgni are called the sign representations. 3. Generalized Robinson-Schensted on the hyperoctahedral group Before explaining a generalization of the Robinson-Schensted algorithm on arbitrary complex reflection groups, we look at the simplest non-trivial example—namely, the group G(2, 1, n), which is the hyperoctahedral group, also known as the signed permutation group. We assume that the reader is already familiar with the standard Robinson-Schensted algorithm on the symmetric group S . Observe that when r = 2, the primitive root of unity for √ n 2π −1 G(r, p, n) is simply ζ = exp 2 = −1. For ease of notation in this section, we omit the ζ in the window notation and simply write an element w ∈ G(2, 1, n) in one-line notation. For 1 0 0 1 0 0 example, instead of writing w = [ζ σ1, ζ σ2, ζ σ3, ζ σ4, ζ σ5, ζ σ6] with σ = 645312 ∈ S6, we write w = 6 4 5 3 1 2, using the underscores to denote the negative entries. We give a generalization of the Robinson-Schensted correspondence between signed permutations and pairs of standard Young bitableaux. This follows from the work of Stanton and White when we restrict their results to the hyperoctahedral setting [5]. 2 Definition 3.1 (RS-correspondence for G(2, 1, n)). Let w = w1w2 ··· wn be a signed per- + mutation in G(2, 1, n). Define w := {wi1 , . . . , wik } to be the set of positive elements in the − order that they appear in w. Define w := {|wik+1 |,..., |win |} to be the set of absolute values of the negative entries in the order that they appear in w. The usual Robinson-Schensted map from Sn to the set of pairs of same-shape standard Young tableaux is denoted

RS : Sn −→ SYTn × SYTn given by RS(π) = (P (π),Q(π)) for π ∈ Sn. From this map we build a generalized Robinson- Schensted map from G(2, 1, n) to the set of pairs of same-shape standard Young bitableaux which we denote RS : G(2, 1, n) −→ SBTn × SBTn given by RS(w) = ( (P +(w),P −(w)) , (Q+(w),Q−(w)) ). In this case, the ﬁrst tableau of each bitableaux pair have shape λ a partition of k, while the second tableau of each bitableaux pair have shape µ a partition of n − k. We construct the four tableaux in the image under the RS-map as follows: RS(w+) = (P +(w),Q+(w)), and RS(w−) = (P −(w),Q−(w)).

Example 3.2. Let w = 6 4 5 3 1 2 be an element of the group G(2, 1, 6), recalling that underscores denote negative entries. Then w+ = {4, 5, 1, 2} and w− = {6, 3}. The standard RS-insertion method on w+ gives the following construction of P +(w) and Q+(w): 1 5 1 2 Insertion tableaux: 4 −→ 4 5 −→ −→ 4 4 5 2 3 2 3 Recording tableaux: 2 −→ 2 3 −→ −→ . 5 5 6 Observe that the recording tableau keeps a record of the position in w from which the particular entry in w+ came. Similarly P −(w) and Q−(w) are given by the following: 3 Insertion tableaux: 6 −→ 6 1 Recording tableaux: 1 −→ . 4 Hence the map RS takes w to the following pair of bitableaux: 1 2 , 3 , 2 3 , 1 . 4 5 6 5 6 4

4. Generalized Robinson-Schensted on G(r, p, n) To generalize the setting given in the previous section, we need some terminology so that we may speak of multitableaux. We write a partition λ of m as a nonincreasing sequence of positive integers (λ1, λ2, . . . , λk) and define |λ| = m.A Young diagram [λ] of λ is a left-justified array of boxes containing λi boxes in its ith row. With the integer r fixed, a multipartition of rank n is an r-tuple λ = (λ0, λ1, . . . , λr−1) of partitions such that Pr−1 i 0 1 r−1 i=0 |λ | = n. The Young diagram [λ] of λ is the r-tuple ([λ ], [λ ],..., [λ ]). We refer to λ as the shape of the diagram [λ] and define |λ| = n. We follow a convention of denoting 3 objects derived from multipartitions in boldface font while writing those derived from single partitions using a normal-weight font. Example 4.1. Consider the multipartition λ = ((2, 1), (1, 1), (2), ∅) of rank 7. Its corresponding Young diagram is the following: [λ] = , , , ∅ .

A standard Young tableaux of shape λ is the Young diagram [λ] together with a labeling of each of its boxes with the elements of Nn in such a way that each number is used exactly once, and the labels of the boxes within each component Young diagram [λi] increase along its rows and down its columns. Remembering that r is fixed, we write SYTn for the set of all standard Young tableaux of rank n whose shape is a multipartition with r components. We are now ready to define a generalized Robinson-Schensted algorithm on complex reflection groups G(r, 1, n). Since G(r, p, n) are just subgroups of G(r, 1, n), it suffices to define the algorithm on G(r, 1, n). As in the hyperoctahedral setting, this follows from the work of Stanton and White [5].

a1 a2 an Definition 4.2 (Generalized RS-correspondence). Let w = [ζ σ1, ζ σ2, . . . , ζ σn] be an (k) element of G(r, 1, n). Define the ordered sets w = (σi | ai = k) for 0 ≤ k < r. Let (k) (k) RS w = (Pk,Qk) be the image of the sequence w under the usual Robinson-Schensted map where Pk is the insertion tableau and Qk is the recording tableau, and define

P(w) = (P0,P1,...,Pr−1) and Q(w) = (Q0,Q1,...,Qr−1).

The Robinson-Schensted map RS : G(r, 1, n) −→ SYTn × SYTn is deﬁned by RS(w) = (P(w), Q(w)). The function RS maps into the set of same-shape pairs of Young tableaux of rank n and is in fact a bijection. Moreover, the image of RS when restricted to a subgroup G(r, p, n) admits a simple description:

Proposition 4.3. The standard Young tableaux pair (P, Q) ∈ RS(G(r, p, n)) if and only if r−1 X k · |shape(Pk)| ≡ 0 (mod p) k=1

a1 a2 an Proof. Write w = [ζ σ1, ζ σ2, . . . , ζ σn] ∈ G(r, p, n), and for each 0 ≤ k ≤ r − 1, set Ak := {i | ai = k}. If RS(w) = (P, Q), then |shape(Pk)| = |Ak|, and hence k · |shape(Pk)| equals P a . Since = Fr−1 A , we conclude that i∈Ak i Nn k=0 k n r−1 r−1 X X X ak = k · |shape(Pk)| = k · |shape(Pk)|. k=1 k=0 k=1 r Finally, since w ∈ G(r, p, n) if and only (ζa1 ζa2 ··· ζan ) p = 1, we conclude that r−1 X w ∈ G(r, p, n) if and only if k · |shape(Pk)| ≡ 0 (mod p), k=1 as claimed. 4 Example 4.4. Consider w = [ζ1 5, ζ0 1, ζ2 3, ζ0 6, ζ2 7, ζ1 4, ζ0 2] ∈ G(4, 1, 7). Then w(0) = (1, 6, 2) =⇒ RS w(0) = 1 2 , 2 4 6 7 w(1) = (5, 4) =⇒ RS w(1) = 4 , 1 5 6 w(2) = (3, 7) =⇒ RS w(2) = 3 7 , 3 5 w(3) = ∅ =⇒ RS w(3) = ( ∅, ∅ ) . Thus we have the following image of w under generalized Robinson-Schensted algorithm: 1 2 4 2 4 1 RS(w) = , , 3 7 , ∅ , , , 3 5 , ∅ . 6 5 7 6 Now consider the inverse of w−1 = [ζ0 2, ζ0 7, ζ2 3, ζ3 6, ζ3 1, ζ0 4, ζ2 5]. We leave it to the reader to verify that −1 2 4 1 1 2 4 RS(w ) = , ∅ , 3 5 , , , ∅ , 3 7 , . 7 6 6 5

In the standard RS-algorithm for the symmetric group, it is well known that for τ ∈ Sn, it follows that RS(τ) = (P,Q) implies RS(τ −1) = (Q, P ). Similarly, it is easily veriﬁed that the generalized RS-algorithm on the hyperoctahedral preserves the same behavior. That is, for w ∈ G(2, 1, n), it follows that RS(w) = (P, Q) implies RS(w−1) = (Q, P). However, this behavior does not hold (as the example above shows) for arbitrary complex reﬂection groups when r > 2. Indeed, the following proposition holds: Proposition 4.5. Let w ∈ G(r, p, n) have corresponding image RS(w) = (P(w), Q(w)), and its inverse w−1 have corresponding image RS(w−1) = (P0(w−1), Q0(w−1)). Then

(P(w), Q(w)) = ((P0,P1,...,Pr−1), (Q0,Q1,...,Qr−1)) , and 0 −1 0 −1 (P (w ), Q (w )) = ((Q0,Qr−1,...,Q1), (P0,Pr−1,...,P1)) .

(k) −1 (r−k) Proof. By comparing the sets w and (w ) for all k, the claim is readily verified. References [1] M. Broué,G. Malle, and R. Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127–190. [2] M. Geck and N. Jacon, Representations of Hecke algebras at roots of unity, Algebra and applications 15, Springer-Verlag London Ltd. (2011), 401pgs. [3] A. Mbirika, T. Pietraho, and W. Silver, On the sign representations for the complex reflection groups G(r, p, n). (Our 2013 submitted version is available by clicking the link below: http://people.uwec.edu/mbirika/paper_Elsevier_submit_version.pdf). [4] G. Shephard and J. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274-304. [5] D. Stanton and D. White, A Schensted algorithm for rim hook tableaux, J. Combin. Theory Ser. A 40 (1985), 211–247.