The modular branching rule for affine Hecke algebras of type A

Abstract

For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani.

Representation theory of symmetric groups and related Hecke algebras

“…In characteristic zero this goes back to a 1908 paper of Schur [178, p. 253]. These branching rules were generalized in various directions, see for example [27,81,82,6,56,41,18,31,132,86,187,193,176,48,8,58]. The graded case is dealt with in [40].…”

“…(6) elements of a standard spanning set of V (Λ) coming from the construction of V (Λ) in terms of a higher level Fock space correspond to the classes of the Specht modules over cyclotomic Hecke algebras [5]; (7) provided F = C, elements of the dual canonical basis in V (Λ) correspond to the classes of the irreducible H Λ d (C, ξ)-modules [3,5]; (8) provided F = C, elements of the canonical basis in V (Λ) correspond to the classes of projective indecomposable H Λ d (C, ξ)-modules [3,5]. One thing which remains unexplained in the picture described above is the role of the quantum group.…”

“…, v N of the natural gl N (C)-module V . Define e ∈ gl N (C) to be the nilpotent matrix of Jordan type λ which maps the basis vector corresponding the ith box to the one immediately to its left, or to zero if there is no such box; in our example, e = e 1,4 + e 2,5 + e 5,7 + e 3,6 + e 6,8 + e 8,9 . Finally, define a Z-grading gl N (C) = r∈Z gl N (C) r on gl N (C) by declaring that e i,j is of degree col(j) − col(i) for each i, j = 1, .…”

CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

“…Then { D µ | µ ∈ K Λ n } is a complete set of pairwise non-isomorphic irreducible H Λ n -modules. The set of multipartitions K Λ n has been determined by Ariki [4]; see also [8,22]. We describe and recover his classification of the irreducible H Λ n -modules in Corollary 3.5.28 below.…”

On the classification of finite dimensional irreducible modules for affine BMW algebras

“…At the end of this section, we recall Goodman's result for 0 < d < r in [14]. In this case, d is the minimal integer such that {e 1 , e 1 x 1 , · · · , e 1 x d 1 } is linear dependent 3 In [29], κ is an arbitrary field. in B r,2 (u).…”