A Difference Formula for Tensor-power Multiplicities
Hongfei Shu, Peng Zhao, Rui-Dong Zhu, Hao Zou
TL;DR
The paper addresses the problem of decomposing tensor powers of a spin-$s$ representation for the $A_r$ Lie algebra by introducing a difference formula that expresses tensor-product multiplicities as signed sums of restricted occupancy coefficients through a shifted Weyl-denominator operator: $\mu_{\lambda} = \mathcal{D}_{R_{A_r}(t)} c_{s,L}(\vec{M})$, where $\lambda_i = M_{i-1}-M_i$, $M_0=2sL$, and $M_{r+1}=0$. The approach connects the Weyl-character expansion $[\chi_{(2s)}(x)]^L = \sum_{\lambda} \mu_{\lambda} \chi_{\lambda}(x)$ with a combinatorial occupancy description $[\chi_{(2s)}(x)]^L = \sum_{\vec{M}} c_{s,L}(\vec{M}) \cdot (\text{monomial in } x_i)$, enabling a direct computation of multiplicities from occupancy data. The results extend to branching rules for subalgebras $\mathfrak{h}$ via $\mu^{\mathfrak{h}}_{\lambda} = \mathcal{D}_{R_{\mathfrak{h}}(t)} c_{s,L}(\vec{M})$ and are further generalized to $A$-type Lie superalgebras $\mathfrak{sl}(m|n)$, employing hook-Schur polynomials $\mathrm{HS}_{\lambda}(x;y)$ and corresponding shifted operators; conjectures are proposed for the superalgebra case with evidence from special cases. The framework provides a novel combinatorial bridge linking tensor-product multiplicities, restricted occupancy, and branching, with potential implications for spin-chain models and representation-theoretic computations. The paper also discusses a dictionary between standard Young diagrams and the occupancy indices, and outlines conjectured formulas for subalgebra branches and superalgebra decompositions, inviting further proof and numerical verification.
Abstract
A novel combinatorial formula is developed for for tensor product multiplicities in representation theory. We introduce a difference formula linking these multiplicities to restricted occupancy coefficients via a shifted operator. This method is extended to derive branching rules for subalgebras and is conjecturally applied to A-type Lie superalgebras.