Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Growth Problems of Quantum Groups

Jensen O'Sullivan, Daniel Tubbenhauer

TL;DR

This work analyzes the asymptotics of $b_n=b_n(T)$, the number of indecomposable summands in tensor powers of tilting modules for quantum groups at a complex root of unity. In type $A_1$ with the defining representation, it yields a sharp, random-walk–driven description built from a half-line walk with a modular constraint, plus a precise leading constant; in general, it proves a universal law: the exponential growth rate is $\beta=\dim_{\mathbb{C}}T$ and the subexponential exponent $\tau$ depends only on the root system (the number of positive roots), making the asymptotics largely module-independent. The SL$_2$ case is developed further, including explicit parity-dependent constants and wall-summand counts, and the results are extended to rank-2 refinements and mixed characteristic via comparisons to characteristic-zero tilting theory and antispherical KL polynomials. The paper thus unifies quantum-root-of-unity behavior with the classical growth picture, providing both universal bounds and precise SL$_2$ instances, and lays groundwork for broader generalizations to other types and characteristics.

Abstract

We study the asymptotic size of decompositions of tensor powers of tilting modules for quantum groups (mostly at a complex root of unity). In type A1 we obtain a sharp result for the number of indecomposable summands, explained by a one dimensional half-line random walk with a periodic congruence constraint. In general type we prove a universal law: the dominant part is governed only by the dimension of the module, while the correction depends only on the root system, so the asymptotic size is largely independent of the specific tilting module.

Growth Problems of Quantum Groups

TL;DR

This work analyzes the asymptotics of , the number of indecomposable summands in tensor powers of tilting modules for quantum groups at a complex root of unity. In type with the defining representation, it yields a sharp, random-walk–driven description built from a half-line walk with a modular constraint, plus a precise leading constant; in general, it proves a universal law: the exponential growth rate is and the subexponential exponent depends only on the root system (the number of positive roots), making the asymptotics largely module-independent. The SL case is developed further, including explicit parity-dependent constants and wall-summand counts, and the results are extended to rank-2 refinements and mixed characteristic via comparisons to characteristic-zero tilting theory and antispherical KL polynomials. The paper thus unifies quantum-root-of-unity behavior with the classical growth picture, providing both universal bounds and precise SL instances, and lays groundwork for broader generalizations to other types and characteristics.

Abstract

We study the asymptotic size of decompositions of tensor powers of tilting modules for quantum groups (mostly at a complex root of unity). In type A1 we obtain a sharp result for the number of indecomposable summands, explained by a one dimensional half-line random walk with a periodic congruence constraint. In general type we prove a universal law: the dominant part is governed only by the dimension of the module, while the correction depends only on the root system, so the asymptotic size is largely independent of the specific tilting module.

Paper Structure

This paper contains 7 sections, 22 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The half-line random walk and a variation
  3. Background
  4. SL2 and quantum SL2
  5. General results
  6. Better bounds in rank 2
  7. SL2 in the mixed case

Key Result

Lemma 1

We have $a_{n,m}=0$ for $m\not\equiv n\bmod 2$, and otherwise $a_{n,m}=\binom{n}{(n-m)/2}-\binom{n}{(n-m)/2-1}$.

Figures (1)

  • Figure 1: The dominant $SL3$ weights $(m,n)\in X^+$, distinguished by color depending on their singularity, arranged in $\ell=5$ scaled alcoves. The alcove containing $I_0$ and the set $\partial\widehat{A}_1$ are illustrated in pink, those containing $I_p$ are illustrated in cyan, and the ones containing $I_b$ form the remaining white collar.

Theorems & Definitions (57)

  • Remark 1
  • Remark 2
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Example 3
  • Lemma 2
  • proof
  • Theorem 1
  • ...and 47 more