[Paper Review] Characters of Feigin-Stoyanovsky's type subspaces of level one modules for affine Lie algebras of types $A_\ell^{(1)}$ and $D_4^{(1)}$

Goran Trupčević|arXiv (Cornell University)|Feb 1, 2010

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper derives character formulas for Feigin-Stoyanovsky’s type subspaces of level one modules in affine Lie algebras of types $A_\ell^{(1)}$ and $D_4^{(1)}$ by leveraging combinatorial bases of monomial vectors satisfying difference and initial conditions. It establishes recurrence relations for these characters and solves them explicitly, yielding closed-form expressions involving $q$-series and partition functions, with key results for $D_4^{(1)}$ derived via decomposition into row and column subconfigurations.

ABSTRACT

We use combinatorial description of bases of Feigin-Stoyanovsky's type subspaces of standard modules of level 1 for affine Lie algebras of types $A_\ell^{(1)}$ and $D_4^{(1)}$ to obtain character formulas. These descriptions naturally lead to systems of recurrence relations for which we also find solutions.

Motivation & Objective

• To derive explicit character formulas for Feigin-Stoyanovsky’s type subspaces of level one standard modules in affine Lie algebras of types $A_\ell^{(1)}$ and $D_4^{(1)}$.
• To establish recurrence relations for the characters of these subspaces based on combinatorial structures of monomial bases.
• To solve these recurrence relations and obtain closed-form expressions involving $q$-series and partition functions.
• To extend known results for $A_\ell^{(1)}$ to the exceptional case $D_4^{(1)}$ using structural decomposition of the root system.

Proposed method

• Utilizes combinatorial descriptions of monomial bases in Feigin-Stoyanovsky’s type subspaces, parameterized by $(k,\ell+1)$-admissible configurations.
• Applies bijections between basis elements and products of partitions to compute graded dimensions and character formulas.
• Decomposes the set of colors $\Gamma$ into row and column components for $D_4^{(1)}$, enabling recursive character computation.
• Derives recurrence relations for characters by analyzing path structures and minimal monomial representatives in the basis.
• Solves the recurrence systems using $q$-series identities and partition-theoretic expressions, particularly involving $(q)_n$-products.
• Employs a recursive decomposition of the root system and associated subalgebras to reduce the $D_4^{(1)}$ case to known $A_2^{(1)}$ and $A_1^{(1)}$ results.

Experimental results

Research questions

• RQ1 How can character formulas for Feigin-Stoyanovsky’s type subspaces of level one modules in $D_4^{(1)}$ be derived using combinatorial basis structures?
• RQ2 What recurrence relations govern the characters of these subspaces, and how can they be solved explicitly?
• RQ3 How does the decomposition of the color set $\Gamma$ into rows and columns facilitate character computation in the $D_4^{(1)}$ case?
• RQ4 What is the role of path structures and minimal monomial representatives in character derivation?
• RQ5 How do the character formulas for $D_4^{(1)}$ relate to known formulas for $A_\ell^{(1)}$ via recursive decomposition?

Key findings

• The character of the Feigin-Stoyanovsky’s type subspace for $D_4^{(1)}$ is given by a sum over $i$ with coefficients involving $q$-series and rational functions of $q$-Pochhammer symbols.
• For $D_4^{(1)}$, the character formula is expressed as a sum over $i$ from $0$ to $n_4 - m' - m''$, with terms involving $q^{f_i(\alpha)}$ and ratios of $q$-Pochhammer symbols.
• The coefficient $d_\gamma(\alpha)$ in the character formula is explicitly given for each $\gamma$, with rational functions capturing combinatorial dependencies.
• The formula for $\chi_{\Gamma';4,\underline{4},\underline{2}}^{\alpha'}$ is derived as a combination of other character terms, reflecting inclusion-exclusion in the basis structure.
• The character formulas for $D_4^{(1)}$ are consistent with known results for $A_2^{(1)}$ when restricted to subconfigurations, validating the decomposition method.
• The solution of the recurrence system yields a closed-form expression for the character that matches earlier results in the literature for $A_\ell^{(1)}$ and extends them to $D_4^{(1)}$.