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On the maximal and minimal degree components of the cocenter of the cyclotomic KLR algebras

Jun Hu, Lei Shi

TL;DR

The paper proves that for cyclotomic KLR algebras over arbitrary fields and types, the cocenter $\mathrm{Tr}(\mathscr{R}_\alpha^\Lambda)$ has nonzero degree components only in even degrees between $0$ and $d_{\Lambda,\alpha}$, extending prior char-0 results to arbitrary characteristic. It provides explicit generators for the maximal and minimal degree components via piecewise dominant sequences, introducing divided-power-like elements $e(\nu)^{(-)}$ in the cocenter and establishing a spanning description involving $Z^\Lambda(\nu)$ and $\mathcal{R}^\Lambda(\nu)$. It further shows that the degree-zero cocenter dimension equals the weight-space dimension $\dim V(\Lambda)_{\Lambda-\alpha}$ and gives a concrete basis using projective covers, complemented by Morita equivalences linking the cyclotomic algebra to its idempotent-corner algebras. These results unify and generalize HS3 and SVV to arbitrary characteristic, with implications for the indecomposability conjecture and the representation theory of $\mathfrak{g}$ through categorification.

Abstract

Let $\mathscr{R}_α^Λ$ be the cyclotomic KLR algebra associated to a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. Shan, Varagnolo and Vasserot show that, when the ground field $K$ has characteristic $0$, the degree $d$ component of the cocenter $Tr(\mathscr{R}_α^Λ)$ is nonzero only if $0\leq d\leq d_{Λ,α}$. In this paper we show that this holds true for arbitrary ground field $K$, arbitrary $\mathfrak{g}$ and arbitrary polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. We generalize our earlier results on the $K$-linear generators of $Tr(\mathscr{R}_α^Λ), Tr(\mathscr{R}_α^Λ)_0, Tr(\mathscr{R}_α^Λ)_{d_{Λ,α}}$ to arbitrary ground field $K$. Moreover, we show that the dimension of the degree $0$ component $Tr(\mathscr{R}_α^Λ)_0$ is always equal to $\dim V(Λ)_{Λ-α}$, where $V(Λ)$ is the integrable highest weight $U(\mathfrak{g})$-module with highest weight $Λ$, and we obtain a basis for $Tr(\mathscr{R}_α^Λ)_0$.

On the maximal and minimal degree components of the cocenter of the cyclotomic KLR algebras

TL;DR

The paper proves that for cyclotomic KLR algebras over arbitrary fields and types, the cocenter has nonzero degree components only in even degrees between and , extending prior char-0 results to arbitrary characteristic. It provides explicit generators for the maximal and minimal degree components via piecewise dominant sequences, introducing divided-power-like elements in the cocenter and establishing a spanning description involving and . It further shows that the degree-zero cocenter dimension equals the weight-space dimension and gives a concrete basis using projective covers, complemented by Morita equivalences linking the cyclotomic algebra to its idempotent-corner algebras. These results unify and generalize HS3 and SVV to arbitrary characteristic, with implications for the indecomposability conjecture and the representation theory of through categorification.

Abstract

Let be the cyclotomic KLR algebra associated to a symmetrizable Kac-Moody Lie algebra and polynomials . Shan, Varagnolo and Vasserot show that, when the ground field has characteristic , the degree component of the cocenter is nonzero only if . In this paper we show that this holds true for arbitrary ground field , arbitrary and arbitrary polynomials . We generalize our earlier results on the -linear generators of to arbitrary ground field . Moreover, we show that the dimension of the degree component is always equal to , where is the integrable highest weight -module with highest weight , and we obtain a basis for .
Paper Structure (6 sections, 18 theorems, 86 equations)

This paper contains 6 sections, 18 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Some nilHecke algebra calculations on cocenters
  4. Explicit generators for the cocenter over arbitrary field
  5. Generators of the cocenter
  6. On the degree zero component of $\mathop{\mathrm{Tr}}\nolimits(\mathscr{R}^\Lambda_\alpha)$

Key Result

Theorem 1.1

Let $K$ be an arbitrary field. Let $\mathscr{R}_{\alpha}^{\Lambda}$ be a cyclotomic KLR algebra over $K$ of arbitrary type. Then $\mathop{\mathrm{Tr}}\nolimits(\mathscr{R}_{\alpha}^{\Lambda})_d\neq 0$ only if $0\leq d\leq d_{\Lambda,\alpha}$ and $d\in 2\mathbb{Z}$. Similarly, $Z(\mathscr{R}_{\alpha}

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6: SVV
  • Definition 3.1
  • ...and 29 more