Journal of Algebraic Combinatorics (2012)
We study the generalized Galois numbers which count flags of length \(r\) in \(N\)-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as \(N\) becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on \(N\) elements and identify their asymptotic limit as the Mahonian inversion statistic when \(r \to \infty\). Finally, we apply our statements to derive further statistical aspects of generalized Rogers–Szegő polynomials, re-interpret the asymptotic behavior of linear \(q\)-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.
Journal of Combinatorial Theory, Series A 118 (2011), 2454–2462
Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras.
Transformation Groups 16 (2011), 1009–1026
We compute the expected degree of a randomly chosen element in a basis of weight vectors in the Demazure module \(V_w(\Lambda)\) of \(\widehat{\mathfrak{sl}}_2\). We obtain en passant a new proof of Sanderson's dimension formula for these Demazure modules.
Journal of Pure and Applied Algebra 214 (2010), 1152–1164
We introduce the notion of a chopped and sliced cone in combinatorial geometry and prove a structure theorem expressing the number of integral points in a slice of such a cone by means of a vector partition function. We observe that this notion applies to weight multiplicities of Kac–Moody algebras and to Clebsch–Gordan coefficients for semisimple Lie algebras. This has algorithmic applications, as we demonstrate computing some explicit examples.
M. Dehmer, M. Drmota, F. Emmert-Streib (Hrsg.), Proceedings of the 2008 international conference on information theory and statistical learning, CSREA Press, 2008, S. 80–86
We explicitly determine quasi-polynomials describing the weight multiplicities of the Lie algebra \(\mathfrak{so}_5(\mathbf{C})\). This information entails immediate complete knowledge of the character of any simple representation as well as the asymptotic behavior of characters.
