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Seminormal basis for the cyclotomic Hecke algebra of type $G(r,p,n)$

Jun Hu, Shixuan Wang

Abstract

The cyclotomic Hecke algebra $H_{r,p,n}$ of type $G(r,p,n)$ (where $r=pd$) can be realized as the $σ$-fixed point subalgebra of certain cyclotomic Hecke algebra $H_{r,n}$ of type $G(r,1,n)$ with some special cyclotomic parameters, where $σ$ is an automorphism of $H_{r,n}$ of order $p$. In this paper we prove a number of rational properties on the $γ$-coefficients arising in the construction of the seminormal basis for the semisimple Hecke algebra $H_{r,n}$. Using these properties, we construct a seminormal basis for the semisimple Hecke algebra $H_{r,p,n}$ in terms of the seminormal basis for the semisimple Hecke algebra $H_{r,n}$. The proof relies on some careful and subtle study on some rational and symmetric properties of some quotients and/or products of $γ$-coefficients of $H_{r,n}$. As applications, we obtain an explicit basis for the center $Z(H_{r,p,n})$ and an explicit basis for the $σ$-twisted $k$-center $Z(H_{r,n})^{(k)}$ of $H_{r,n}$ for each $k\in\mathbb{Z}/p\mathbb{Z}$.

Seminormal basis for the cyclotomic Hecke algebra of type $G(r,p,n)$

Abstract

The cyclotomic Hecke algebra of type (where ) can be realized as the -fixed point subalgebra of certain cyclotomic Hecke algebra of type with some special cyclotomic parameters, where is an automorphism of of order . In this paper we prove a number of rational properties on the -coefficients arising in the construction of the seminormal basis for the semisimple Hecke algebra . Using these properties, we construct a seminormal basis for the semisimple Hecke algebra in terms of the seminormal basis for the semisimple Hecke algebra . The proof relies on some careful and subtle study on some rational and symmetric properties of some quotients and/or products of -coefficients of . As applications, we obtain an explicit basis for the center and an explicit basis for the -twisted -center of for each .

Paper Structure

This paper contains 5 sections, 42 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some rational properties of the $\gamma$-coefficients of $\mathscr{H}_{r,n}$
  4. Seminormal bases for $\mathscr{H}_{r,p,n}$
  5. The $\sigma$-twisted $k$-centers of $\mathscr{H}_{r,n}$

Key Result

Theorem 1.2

Suppose $\mathscr{H}_{r,n}$ is semisimple. Let ${\boldsymbol\lambda}\in\mathscr{P}_{r,n}$ and $\mathfrak{t}\in\mathop{\mathrm{Std}}\nolimits({\boldsymbol\lambda})$. Let $0\leq k,l\leq p_{{\boldsymbol\lambda}}$. Then we have where $o_{\boldsymbol\lambda}, p_{\boldsymbol\lambda}$ are defined in (olamb), $\mathfrak{t}\langle j\rangle$ is defined in (langlerangle).

Theorems & Definitions (87)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 3.2: Ma04
  • Lemma 3.4
  • Lemma 3.5
  • ...and 77 more