Graded Cartan determinants of the symmetric groups

Abstract

Abstract. We give the graded Cartan determinants of the symmetric groups. Based on it, we propose a gradation of Hill's conjecture which is equivalent to Külshammer-OlssonRobinson's conjecture on the generalized Cartan invariants of the symmetric groups.

On graded Cartan invariants of symmetric groups and Hecke algebras

“…The following result is similar to [Tsu,Proposition 2.3] and is proved by essentially the same argument as that given in [BK1,§5]. We include a proof for clarity.…”

“…λ are defined as in Proposition 2.17. Moreover, by an identity in the proof of [Tsu,Theorem 4.4] 1 (together with the definition of ·, · QSh ), we have…”

“…The equivalence class of C v d (X) under does not depend on the choice of w ∈ W [Tsu, Proposition 3.18]. The following is implicit in the proof of [Tsu,Theorem 4.4]. For convenience, we give a proof.…”

“…, ℓ}. As in the proof of [Tsu,Theorem 4.4], V (Λ 0 ) A Λ0−dδ can be regarded as an A -lattice of the polynomial ring k[h i,−r | i ∈ I, r ≥ 1]. More precisely, defining new variables y…”

Characters of Integrable Highest Weight Modules over a Quantum Group

“…When g is finite-dimensional, this is shown in Andersen, Polo and Wen [1]. When g is affine, this is known in certain specific cases (see Chari and Jing [2], Tsuchioka [15]).…”