[Paper Review] Character formulas for Feigin-Stoyanovsky's type subspaces of standard $\mathfrak{sl}(3, \mathbb{C})^{\widetilde{}}$-modules

Miroslav Jerkovi ' c|arXiv (Cornell University)|May 15, 2011

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper derives new fermionic-type character formulas for Feigin-Stoyanovsky’s type subspaces of standard $ω(3,\mathbb{C})^\sim$-modules at arbitrary integer level by solving a system of recurrence relations for formal characters. Using combinatorial techniques and recursive transformations, the authors establish a closed-form solution that generalizes known results and reduces to previously established formulas under specialization.

ABSTRACT

Exact sequences of Feigin-Stoyanovsky's type subspaces for affine Lie algebra $\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}}$ lead to systems of recurrence relations for formal characters of those subspaces. By solving the corresponding system for $\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}$, we obtain a new family of character formulas for all Feigin-Stoyanovsky's type subspaces at general level.

Motivation & Objective

• To derive exact character formulas for Feigin-Stoyanovsky’s type subspaces of standard $ω(3,\mathbb{C})^\sim$-modules at general integer level.
• To solve a system of recurrence relations for formal characters of these subspaces using combinatorial and algebraic techniques.
• To generalize existing character formulas by constructing a new family of fermionic-type expressions.
• To show that the derived formulas reduce to known results under appropriate specializations.
• To provide a systematic method for computing characters of these subspaces using recursive transformations and generating functions.

Proposed method

• Formal characters of Feigin-Stoyanovsky’s type subspaces are defined using the action of the $ω_{1}$-subalgebra of the affine Lie algebra $ω(3,\mathbb{C})^\sim$.
• A system of recurrence relations for the characters is derived from exact sequences of subspaces at general level.
• The recurrence system is solved via a sequence of combinatorial transformations involving summation index shifts and generating function identities.
• Key identities involve $q$-series and $q$-Pochhammer symbols, particularly through the use of $l_p^2(q)$, $\delta_p^2(q)$, and $N_{k_0,k_1,k_2}(q)$ functions.
• Recursive transformations are applied to sums over partitions with constraints, using bijections between index sets to re-express generating functions.
• The proof relies on technical lemmas establishing equivalence between transformed sums and original character expressions, culminating in Theorem 3.11.

Experimental results

Research questions

• RQ1 Can a closed-form fermionic-type character formula be derived for Feigin-Stoyanovsky’s type subspaces of $ω(3,\mathbb{C})^\sim$-modules at general integer level?
• RQ2 How do the recurrence relations for formal characters of these subspaces relate to known Rogers-Selberg-type recursions?
• RQ3 What is the precise structure of the combinatorial basis of admissible monomial vectors that underlies the character formula?
• RQ4 Under what specializations do the new character formulas reduce to previously established results in the literature?
• RQ5 Can the recursive system for characters be solved using a transformation-based method analogous to Andrews’ approach to Rogers-Selberg identities?

Key findings

• The paper establishes a new family of fermionic-type character formulas for all Feigin-Stoyanovsky’s type subspaces of standard $ω(3,\mathbb{C})^\sim$-modules at general integer level.
• The solution to the recurrence system is expressed through a sum over partitions with specific constraints, involving $q$-series and $q$-Pochhammer products.
• The character formulas are shown to reduce to known results from [FJMMT, FJMMT2] under appropriate specializations, confirming consistency with prior work.
• The proof relies on a chain of lemmas transforming sums via index shifts and bijections, ultimately proving equivalence between transformed and original expressions.
• The method successfully generalizes the approach used in [A] for Rogers-Selberg recursions to the case of $ω(3,\mathbb{C})^\sim$ subspaces.
• The final character formula is given in Theorem 3.11 and is verified through a sequence of technical lemmas on generating functions and partition identities.