Abstract
We derive a formula for the entries in the change-of-basis matrix between Young's seminormal and natural representations of the symmetric group. These entries are determined as sums over weighted paths in the weak Bruhat graph on standard tableaux, and we show that they can be computed recursively as the weighted sum of at most two previously-computed entries in the matrix. We generalize our results to work for affine Hecke algebras, Ariki-Koike algebras, Iwahori-Hecke algebras, and complex reflection groups given by the wreath product of a finite cyclic group with the symmetric group.
Highlights
Young [You] defined three bases of the complex irreducible symmetric group module Sλn for each integer partition λ n
Using this fact we extend our result to give the transition matrix between the natural and seminormal representations, in the semisimple cases, of the following algebras: (1) affine Hecke algebras of type A; (2) cyclotomic Hecke algebras (i.e., Ariki-Koike algebras), (3) Iwahori-Hecke algebras of type A and B; and (4) the group algebras of the complex reflection groups Gr,n = Zr Sn
In Corollary 4.6 and Remark 5.3, we show that the transition matrix entries can be computed recursively in such a way that each entry is the weighted sum of at most two entries in a previous column
Summary
Young [You] defined three bases of the complex irreducible symmetric group module Sλn for each integer partition λ n. We chose to organize this paper with the symmetric group first in order to get to this fundamental result in an accessible way without the extra notation needed to describe Hecke algebra representations For each of these algebras, the transition matrix is upper triangular under any linear the electronic journal of combinatorics 28(3) (2021), #P3.15 ordering on tableaux that extends Bruhat order. For the cyclotomic Hecke algebras and complex reflection groups Gr,n, skew shapes are identified with r-tuples of partitions having a total of n boxes. In this case, the transition matrix decomposes as the direct sum of identical copies of an r-fold tensor product of symmetric group transition matrices (see Corollary 6.6 and Example 6.7).
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