A length function for the complex reflection

Transcription

1 A length function for the complex reflection group G(r, r, n) Eli Bagno and Mordechai Novick SLC 78, March 28, 2017

2 General Definitions S n is the symmetric group on {1,..., n}. Z r is the cyclic group of order r. ζ r is the primitive r th root of unity.

3 Complex reflection groups G(r, n) = group of all matrices π = (σ, k), where: σ = a 1 a n S n. k = (k 1,..., k n ) Z n r. (k-vector) π = (σ, k) is the n n monomial matrix with non-zero entries in the (a i, i) positions. ζ k i r Example (n = 3, r = 4) π(312, (1, 3, 3)) = 0 i i i 0 0

4 For p r, G(r, p, n) is the subgroup of G(r, n) consisting of matrices (σ, k) satisfying n (ζ k i r ) r p = 1. i=1 Hence G(r, r, n) is the group of such matrices satisfying: n (ζ k i r ) = 1 i=1

5 One-line notation We denote an element of G(r, p, n) in a more concise manner: (σ, k) = a k 1 1 akn n for σ = a 1 a n and k = (k 1,..., k n ). Example π(312, (1, 3, 3)) =

6 Our goal Various sets of generators have been defined for complex reflection groups but (as far as we know), no length function has been formulated. We provide such a function for the case of G(r, r, n) with a specific choice of generating set proposed by Shi.

7 Shi s Generators for G(r, r, n) For each i {1,..., n 1} let s i = (i, i + 1) be the familiar adjacent transpositions generating S n. Theorem Define t 0 = (1 r 1, n 1 ). The set {t 0, s 1,..., s n 1 } generates G(r, r, n).

8 Example of generators acting from the right Applying s 1 from the right: Applying t 0 from the right: π = π = Remark Places are exchanged, the k vector is not preserved.

9 Example of generators acting from the left Applying s 1 from the left: Applying t 0 from the left: π = π = Remark Numbers are exchanged and the k-vector is preserved.

10 The affine Weyl group S n is defined as follows: S n = {w : Z Z w(i+n) = w(i)+n, i {1,..., n}, n ( ) n + 1 w(i) = }. 2 i=1

11 Each affine permutation can be written in integer window notation in the form: π = (π(1),..., π(n)) = (b 1,..., b n ). By writing b i = n k i + a i, we can use the residue window notation: π = a k 1 1 akn n. where {a 1,..., a n } = {1,..., n}.

12 Generators for the affine group For each i {1,..., n 1} let s i = (i, i + 1) be the known adjacent transpositions generating S n. Define s 0 = (1, n 1 ).

13 Theorem Let π = a k 1 1 akn n S n. Then l(π) = 1 i<j n a i <a j k j k i + 1 i<j n a i >a j k j k i 1 Example If π = then: l(π) = 1 ( 1) ( 1) ( 1) = 5

14 Another presentation of S n Each affine permutation π = a k 1 1 akn n can also be written as a monomial matrix: { 0 i σ(j) M π = (m ij ) = x k i i = σ(j) Example (n = 4) π = = 0 x x 0 x x 1 0

15 Mapping S n to G(r, r, n) Shi defines a homomorphism η : S n G(r, r, n) by substituting a primitive r-th root of unity ζ r in place of x. He tried to adapt his length function for the affine groups to the case of G(r, r, n) but did not obtain a closed formula. Here we provide such a formula.

16 Difficulties in adapting Shi s formula In G(r, r, n) each element does not have a uniquely defined k- vector, as adding a multiple of r to any k i does not change π as an element of G(r, r, n). Example The permutations and represent the same element of G(5, 5, 4).

17 The normal form Definition A permutation (p, k 0 ) G(r, r, n) is said to be in normal form if the following conditions are met: n 1 ki 0 = 0 i=1 2 max(k 0 ) min(k 0 ) r 3 If there exist i < j such that k 0 j k 0 i = r then k0 j k 0 i = r. If (p, k 0 ) is in normal form and is equivalent to (p, k) then we say that (p, k 0 ) is a normal form of (p, k).

18 Example The normal form of G(7, 7, 4) is Theorem For each π G(r, r, n) a normal form exists and is unique. Shi s length function, when applied to all representatives of a permutation in G(r, r, n), attains its minimum on the normal form representative.

19 Decomposition Into Right Cosets of S n Let π = (k, σ) G(r, r, n). As we have seen, for each generator τ of S n, π and τπ have the same k-vector. Hence, it is natural and straightforward to decompose G(r, r, n) into right cosets. Each right coset has a unique representative π = (k, σ) which has minimal length. This leads us to a new length function for G(r, r, n).

20 The length function for G(r, r, n) Let π = a k 1 1 akn n G(r, r, n). Write π = u σ where u S n and σ is the minimal length representative. Then: Theorem l(π) = where 1 i<j n and (as usual) k j k i noninv(k) + inv(u) noninv(k) = #{(i, j) i < j, k(i) < k(j)} inv(u) = #{(i, j) i < j, u(i) > u(j)}.

21 Length Example Let π = G(4, 4, 4). Then σ = , and u = π σ 1 = Hence: 1 i<j n k j k i = ( 2) + 1 ( 2) = 10 And: while noninv(k) = 3 inv(u) = 5 so that l(π) = = 12

22 Finding the minimal-length representative The minimal-length element σ = a k 1 1 akn n G(r, r, n) for the k-vector (k 1,..., k n ) (abbreviated a 1 a 2 a n S n ) is the unique one with the following property: a i < a j iff: Example k(i) > k(j), or k(i) = k(j) and i < j If k = ( 2, 1, 1, 1, 2, 1) then σ =

23 Open question: What is the generating function? Let G r,r,n (q) = q l(π). π G r,r,n From the coset decomposition it is clear that G r,r,n (q) has [n] q! as a factor. Example G 4,4,4 (q) = [4] q!(1+2q 2 +3q 3 +4q 4 +5q 5 +7q 6 +8q 7 +10q 8 +12q 9 +7q 10 +3q 11 ) G 6,6,3 (q) = [3] q!(1+q+2q 2 +2q 3 +3q 4 +3q 5 +4q 6 +4q 7 +5q 8 +5q 9 +6q 10 )

24 A possible direction... There is a bijection between left cosets of S n in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)). In B-B, each partition is the inversion table of the corresponding left coset (i.e., of its ascending minimal-length representative). The bijection in E-E maps each left coset to the conjugate of its inversion table.

25 A possible direction... There is a bijection between left cosets of S n in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)). In B-B, each partition is the inversion table of the corresponding left coset (i.e., of its ascending minimal-length representative). The bijection in E-E maps each left coset to the conjugate of its inversion table.

26 A possible direction... There is a bijection between left cosets of S n in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)). In B-B, each partition is the inversion table of the corresponding left coset (i.e., of its ascending minimal-length representative). The bijection in E-E maps each left coset to the conjugate of its inversion table.

27 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).

28 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).

29 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).

30 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).

31 Thank you!!