[Paper Review] A vector partition function for the multiplicities of sl_k(C)

Sara Billey, Victor Guillemin+1 more|ArXiv.org|Jul 16, 2003

Algebraic structures and combinatorial models3 references13 citations

TL;DR

This paper expresses the weight multiplicity function for the Lie algebra $ \mathfrak{sl}_k\mathbb{C}$ as a single partition function using Gelfand-Tsetlin diagrams, enabling the application of convex and integer geometry to analyze its structure. The key result is that the weight multiplicity function and the Duistermaat-Heckman measure induce identical partitions of the permutahedron into domains of polynomiality, with explicit connections to Kostant's formula and factorization patterns in boundary weight polynomials.

ABSTRACT

We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat-Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl_4(C) (A_3).

Motivation & Objective

• To express weight multiplicities for $\mathfrak{sl}_k\mathbb{C}$ as a single partition function using Gelfand-Tsetlin diagrams.
• To relate the structure of weight multiplicity diagrams to symplectic geometry via the Duistermaat-Heckman function.
• To explain the appearance of linear factors in weight polynomials on the boundary of the permutahedron.
• To prove that the partition of the permutahedron into polynomiality domains for weight multiplicities matches that of the Duistermaat-Heckman function.
• To provide an elementary proof for $\mathfrak{sl}_4\mathbb{C}$ of the equivalence between the two partitions.

Proposed method

• Represent weight multiplicities via a single partition function $\phi_{E_k}(B_k(\lambda, \beta))$, reducing the counting problem to integer points in polytopes.
• Use Gelfand-Tsetlin diagrams to model the weight space decomposition and derive the matrix formulation of the partition function.
• Apply results from integer programming and chamber complexes to analyze polynomiality domains of the multiplicity function.
• Relate the multiplicity function to the Duistermaat-Heckman measure through the 'Quantization Commutes with Reduction' theorem.
• Use Kostant arrangements to study polynomial factorization and jumps between adjacent regions in the chamber complex.
• Verify factorization patterns via explicit computation in $\mathfrak{sl}_4\mathbb{C}$ and analyze central and boundary domains.

Experimental results

Research questions

• RQ1 How can the weight multiplicity function for $\mathfrak{sl}_k\mathbb{C}$ be expressed as a single partition function using Gelfand-Tsetlin diagrams?
• RQ2 What is the relationship between the chamber complex of the weight multiplicity function and the Duistermaat-Heckman function on the permutahedron?
• RQ3 Why do weight polynomials in boundary regions of the weight diagram exhibit multiple linear factors?
• RQ4 How do jumps between adjacent polynomiality domains relate to factorization patterns in the weight polynomials?
• RQ5 What is the structure of the central domain in the permutahedron for generic $\lambda$, and how does it affect the multiplicity function?

Key findings

• The weight multiplicity function for $\mathfrak{sl}_k\mathbb{C}$ is expressible as a single partition function $\phi_{E_k}(B_k(\lambda, \beta))$, enabling uniform analysis across all weights.
• The partition of the permutahedron into domains of polynomiality for the weight multiplicity function coincides exactly with that of the Duistermaat-Heckman function, with walls defined by convex hulls of $\mathrm{conv}(W\cdot\sigma(\lambda))$.
• For a domain $R$ with a facet on the permutahedron boundary normal to $\theta(\omega_j)$, the weight polynomial $p_R$ is divisible by $j(k-j)-1$ linear factors of the form $\gamma + i$ or $\gamma - i$, where $\gamma$ is the hyperplane equation.
• The Ehrhart quasipolynomial of Gelfand-Tsetlin polytopes is always a polynomial, despite the polytopes not being integral in general, due to the polynomiality property of the multiplicity function.
• For $\mathfrak{sl}_4\mathbb{C}$, the central domain is either a tetrahedron or a truncated cube depending on $\lambda$, and the multiplicity function is constant over the central domain when $\lambda_1 < -\lambda_4$, corresponding to lacunary domains.
• In the case $\lambda_1 = \lambda_2$ or $\lambda_3 = \lambda_4$, the weight polynomials are quadratic and exhibit two parallel linear factors in jumps between adjacent regions, with no extra factors beyond the expected count.