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Graded Character Formula for Fusion Products of Irreducible Modules and Littlewood-Richardson Coefficients in Type $A_2$

Tanusree Khandai, Shushma Rani

Abstract

We investigate a specific class of CV modules for $\mathfrak{sl}_3$ and establish an exact sequence for these modules. Utilizing dimension arguments, we demonstrate that this module is isomorphic to the fusion product of irreducible modules, thereby offering a new proof of the conjecture regarding the independence of fusion products from parameters. By analyzing the filtration of the kernel within the exact sequence, we derive the graded character formula for fusion product modules. Moreover, we leverage the graded character to deduce the algebraic characterization of the Littlewood-Richardson (LR) coefficients and present an alternative proof of the saturation theorem in type $A_2$.

Graded Character Formula for Fusion Products of Irreducible Modules and Littlewood-Richardson Coefficients in Type $A_2$

Abstract

We investigate a specific class of CV modules for and establish an exact sequence for these modules. Utilizing dimension arguments, we demonstrate that this module is isomorphic to the fusion product of irreducible modules, thereby offering a new proof of the conjecture regarding the independence of fusion products from parameters. By analyzing the filtration of the kernel within the exact sequence, we derive the graded character formula for fusion product modules. Moreover, we leverage the graded character to deduce the algebraic characterization of the Littlewood-Richardson (LR) coefficients and present an alternative proof of the saturation theorem in type .
Paper Structure (19 sections, 17 theorems, 159 equations)

This paper contains 19 sections, 17 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Chari Venkatesh Modules
  4. The set $P^+{(\lambda,k)}$ and module $\mathcal{F}_{{\hbox{\boldmath $\lambda$}}}$ for ${\hbox{\boldmath $\lambda$}} \in P^+(\lambda,k)$
  5. The CV module ${\mathcal{F}}_{\lambda,\mu}$ for $\mathfrak{sl}_3[t]$
  6. Filtration of $\operatorname{Ker} \phi(\lambda, \mu)$
  7. Proof of Theorem \ref{['filtration']}
  8. Proof of Proposition \ref{['l*m.dim']}(i).
  9. Proof of Proposition \ref{['l*m.dim']}(ii)
  10. Proof of Proposition \ref{['l*m.dim']}(iii)
  11. Proof of Theorem \ref{['filtration']}
  12. Graded Character of Fusion product modules $\mathcal{F}_{\lambda,\mu}$
  13. Proof of Theorem \ref{['littlewood coeff']}(i):
  14. Proof of Theorem \ref{['littlewood coeff']}
  15. Proof of Theorem \ref{['littlewood coeff']}(iii).
  16. ...and 4 more sections

Key Result

Lemma 1

Given $s\in \mathbb N$, $r\in \mathbb Z_+$ and $\alpha\in R^+$, we have where $P_\alpha(u)_k$ denotes the coefficient of $u^k$ in the power series $H_\alpha(u)$ and $(y)^{(s)} := \frac{y^s}{s!}$ for $y \in \mathfrak g[t]$.

Theorems & Definitions (21)

  • Lemma
  • Lemma
  • Remark
  • Definition
  • Lemma
  • Lemma
  • Lemma
  • Remark
  • Definition
  • Theorem
  • ...and 11 more