Why would multiplicities be log-concave ?

A vector partition function for the multiplicities of sl_k(C)

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 symplect…

On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar S$ for representation $R=[3,1]$ with multiplicities

Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of arXiv:1612.00422, we obtain the exclusive Racah matrix $\bar S$ for the currently-front-line case of representation $R=[3,1]$ with non-trivial multiplicities, where it is actually operator valued, i.e. depends on the choice of basises in the intertwiner spaces. E…

抛物 Verma 模的特征标与受限 Kostant 分拆函数的对数凹性

Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions

本文证明 A 型半单李代数上所有抛物 Verma 模的移位特征标为反归一化 Lorentzian，并通过无环多重图的受限 Kostant 分拆函数给出机制，同时界定该性质的极大性并部分解决相关猜想。