A Hecke Algebra of (Z/rZ)Sn and Construction of Its Irreducible Representations

Dyck tilings and the homogeneous Garnir relations for graded Specht modules

“…Proof of Proposition 5.5 To see that μ satisfies exactly one of the four conditions, let l denote the lowest node in column j which is not a node of μ, and consider whether SE(l) and SW(l) lie in μ; for the purposes of this argument we regard all nodes of the form (a, 0) or (0, b) as lying in μ. If both SE(l) and SW(l) lie in μ, then l is an addable node and we are in case (1). If neither lies in μ, then S(l) is a removable node, and we are in case (2).…”

“…M ( f,g) has a basis indexed by the set of minimal left coset representatives of S f ×S g in S k ; this set is in one-to-one correspondence with the set P f,g of partitions λ for which λ 1 f and λ 1 g; given λ ∈ P f,g , we write the corresponding basis element as t λ m, where…”

“…, c n 0, i ∈ (Z/eZ) n is a basis for R n . The quiver Hecke algebra has attracted considerable attention in recent years, thanks to the astonishing result of Brundan and Kleshchev [3, Main Theorem] that when p does not divide e, a certain finite-dimensional 'cyclotomic' quotient of R n is isomorphic to the cyclotomic Hecke algebra of type A (as introduced by Ariki-Koike [1] and Broué-Malle [2]) defined at a primitive eth root of unity in F; when e = p, there is a corresponding isomorphism to the degenerate cyclotomic Hecke algebra (which includes the group algebra FS n as a special case). This in particular shows that these Hecke algebras (which include the group algebra of the symmetric group) are nontrivially Z-graded, and has initiated the study of the graded representation theory of these algebras.…”

Generalized Blob Algebras and Alcove Geometry

“…In our case there is effectively no ordinary Weyl group and no dominant region; that is to say, the placement of all hyperplanes is controlled by parameters of the algebra. Thus the weight space for the blob algebra, just as for sl 2 , is the Euclidean space associated to the A 1 Coxeter system -that is, it is effectively R (see [57,Section 6]). Now, however, (compare sl 2 ) all integral weights are dominant; that is to say, simple modules may be indexed by Z.…”

“…It is customary to study them through certain families of quotient algebras, among which the usual choice is the cyclotomic Hecke algebras [2,10]. These quotients, though finite-dimensional, are still complicated, and complete knowledge of their representation theory remains a significant challenge.…”