[Paper Review] Defining relations of fusion products and Schur positivity

Katsuyuki Naoi|arXiv (Cornell University)|Apr 1, 2015

Advanced Combinatorial Mathematics12 references3 citations

TL;DR

This paper establishes defining relations for fusion products of $σ$-twisted $σ$-stable modules over the current algebra $σffiα[t]$, generalizing earlier results for $σ=1$. It proves that such fusion products are isomorphic to modules defined by generators and relations, and derives a surjective homomorphism between fusion products when partition dominance holds—offering a current algebra analog of Schur positivity.

ABSTRACT

In this note we give defining relations of an $\mathfrak{sl}_{n+1}[t]$-module defined by the fusion product of simple $\mathfrak{sl}_{n+1}$-modules whose highest weights are multiples of a given fundamental weight. From this result we obtain a surjective homomorphism between two fusion products, which can be considered as a current algebra analog of Schur positivity.

Motivation & Objective

• To determine explicit defining relations for fusion products of $σ$-twisted $σ$-stable modules over $σffiα[t]$.
• To generalize the Schur positivity phenomenon from finite-dimensional $σffiα$-modules to current algebra modules.
• To establish a surjective homomorphism between fusion products $V_m(\bm{\ell})$ and $V_m(\bm{r})$ when $\ell_i + \cdots + \ell_p \geq r_i + \cdots + r_p$ for all $i$
• To provide a current algebra analog of the classical Clebsch-Gordan decomposition and Schur positivity in tensor products.

Proposed method

• Realizes the fusion product $V_m(\bm{\ell})$ as a ${\mathfrak{g}}[t]$-submodule inside a module over the affine Lie algebra $\widehat{{\mathfrak{g}}}$.
• Uses the realization from [Nao12] to recursively determine defining relations via the action of $\tau$-shifted root vectors.
• Applies the method of [Nao13] to analyze annihilators of cyclic vectors under $\widehat{{\mathfrak{n}}}_+$, using $\tau$-conjugation to relate ideals in $U(\widehat{{\mathfrak{n}}}_+)$.
• Employs the $\tau$-action to relate generators $f_\alpha \otimes t^s$ and $e_\alpha \otimes t$ to the twisted structure, particularly for roots with $\langle h_\alpha, \varpi_m \rangle = 1$.
• Uses the $\sigma$-twisted action to define the module $V(q,0)$ as a tensor product involving $\mathbb{C}_{(\ell_q - \ell_{q+1})(\varpi_m + \Lambda_0)}$ and a shifted module.
• Applies the automorphism $\tau \circ \Phi_\sigma$ to relate $e_\alpha^{(s-r)}(f_\alpha \otimes t)^{(s)}$ to the generators ${}_1f_\alpha(r,s)$, proving inclusion relations in the annihilator ideal.

Experimental results

Research questions

• RQ1 What are the defining relations of the fusion product $V_m(\bm{\ell})$ for $\mathfrak{sl}_{n+1}[t]$-modules with highest weights being multiples of a fundamental weight?
• RQ2 Can a surjective ${\mathfrak{g}}[t]$-module homomorphism be constructed between two such fusion products when the partition dominance condition holds?
• RQ3 Is there a current algebra analog of Schur positivity, where the difference of characters is a non-negative sum of Schur functions?
• RQ4 How does the $\sigma$-twisted structure of the module interact with the current algebra action to determine the defining relations?
• RQ5 Can the annihilator of the cyclic vector in the affine realization be explicitly described in terms of generators and relations?

Key findings

• The fusion product $V_m(\bm{\ell})$ is isomorphic to the ${\mathfrak{g}}[t]$-module generated by a vector $v$ satisfying specific relations involving $\mathfrak{n}_+[t]$, $h \otimes t^s$, and $f_\alpha \otimes \mathbb{C}[t]$.
• The defining relations include $f_\alpha^{L_1+1}v = 0$ for positive roots $\alpha$ with $\langle h_\alpha, \varpi_m \rangle = 1$, and $ (e_\alpha \otimes t)^s f_\alpha^{r+s}v = 0 $ under the condition $s + r \geq 1 + kr + L_{k+1}$.
• A surjective ${\mathfrak{g}}[t]$-module homomorphism exists from $V_m(\bm{\ell})$ onto $V_m(\bm{r})$ whenever $\ell_i + \cdots + \ell_p \geq r_i + \cdots + r_p$ for all $1 \leq i \leq p$, generalizing Schur positivity to the current algebra setting.
• The annihilator of the cyclic vector $v_{q,0}$ in the affine realization is shown to equal $\mathcal{I}(q,0)$, confirming the defining relations via $\tau$-conjugation.
• The proof relies on showing that $\tau(\mathcal{I}(q+1, \ell(\sigma))) = \mathcal{I}(q,0)$, which matches the required annihilator ideal.
• The result confirms that the fusion product structure is completely determined by the partition $\bm{\ell}$ and the weight $\varpi_m$, with the relations encoding the dominance order.