Some Graded Representations of the Complex Reflection Groups

Abstract

Let C[X, Y ] denote the ring of polynomials with complex coefficients in the variables X=[x 1 , ..., x n ] and Y=[ y 1 , ..., y n ], let S n denote the symmetric group of order n!, let C m denote the cyclic group C m =[e 2?ijÂm : 0 j m&1] of order m, let H k denote the subgroup of order k of C m , and let G n, m =C m " S n (the wreath product of C m with S n

Multiple Left Regular Representations Generated by Alternants

“…K Allen [1] has constructed bases of the modules M %+\ pertaining to this proposition when 0 a<b, including a different decomposition of (3.2) than what we presented. …”

Bitableau bases for Garsia–Haiman modules of hollow type

“…A proof can be found in [4] (see Eqs. (6.5) and (6.6) in Theorem 6.2), however, there is a sign missing, so we include a proof here.…”

“…We note here that Garsia-Haiman modules corresponding to specific classes of lattice diagrams have been studied elsewhere. Periodic Garsia-Haiman modules were considered by the first author [4] and (in one variable) by H. Morita and H.-F. Yamada [5], R. Stanley [6] and J. Stembridge [7]. Dense Garsia-Haiman modules were investigated by the first author in [8].…”

“…(See[4, Theorem 5.4 and Corollary 5.5].) Suppose g is a map from CS n,A to CS n,A such that for all W ∈ CS n,A , W and g(W ) have the same standardization…”

“…T ,U = (2, 3)(4,5). We consider the expansion ofh 1,1 (∂ X )h 2,1 (∂ Y )[T , C] per (∂ X , ∂ Y )Δ γ = [T , C] per (∂ X , ∂ Y )h 1,1 (∂ X )h 2,1 (∂ Y )Δ γ = [T , C] per (∂ X , ∂ Y ) c β Δ L[β] + δ> cont β c δ Δ L[δ] = c β [T , C] per (∂ X , ∂ Y ) Δ L[β] + δ> cont β c δ [T , C] per (∂ X , ∂ Y )Δ L[δ] ,…”