Related papers: Littlewood--Richardson coefficients and integrable…
Motivated by string theory connection, a covariant procedure for perturbative calculation of the partition function of the two-dimensional generalized $\sigma$-model is considered. The importance of a consistent regularization of the…
The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$.…
We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{\mu\le\lambda} g^{(k)}_{\mu}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by…
We present a library of formalized results around symmetric functions and the character theory of symmetric groups. Written in Coq/Rocq and based on the Mathematical Components library, it covers a large part of the contents of a graduate…
The Newell-Littlewood numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood-Richardson coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian…
We develop structural insights into the Littlewood-Richardson graph, whose number of vertices equals the Littlewood-Richardson coefficient c({\lambda},{\mu},{\nu}) for given partitions {\lambda}, {\mu}, and {\nu}. This graph was first…
We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both…
In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an…
Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the…
Let R\_{0,n} be the Clifford algebra of the antieuclidean vector space of dimension n. The aim is to built a function theory analogous to the one in the C case. In the latter case, the product of two holomorphic functions is holomorphic,…
Given dominant integral weights $\lambda, \mu, \nu$ of a finite-dimensional simple Lie algebra $\mathfrak{g}$ and an element $w$ of its Weyl group, the refined tensor product multiplicity $c_{\lambda \mu}^\nu(w)$ is the multiplicity of the…
Suppose \mu and \mu' are two partitions. We will let \mu \oplus \mu' denote the "direct sum" of the partitions, defined as the sorted partition made of the parts of $\mu$ and $\mu'$. In this paper, we define a summation operation on two…
Cauchy summation formula plays a central role in application of character calculus to many problems, from AGT-implied Nekrasov decomposition of conformal blocks to topological-vertex decompositions of link invariants. We briefly review the…
Let the formal power series f in d variables with coefficients in an arbitrary field be a symmetric function decomposed as a series of Schur functions, and let f be a rational function whose denominator is a product of binomials of the form…
We give an algorithm for working out the indecomposable direct summands in a Krull--Schmidt decomposition of a tensor product of two simple modules for G=SL_3 in characteristics 2 and 3. It is shown that there is a finite family of modules…
Oberdieck and Pandharipande conjectured that the curve counting invariants of $S\times E$, the product of a $K3$ surface and an elliptic curve, is given by minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive…
An introduction is given to the Littlewood-Richardson rule, and various combinatorial constructions related to it. We present a proof based on tableau switching, dual equivalence, and coplactic operations. We conclude with a section…
We make progress towards understanding the structure of Littlewood-Richardson coefficients $g_{\lambda,\mu}^{\nu}$ for products of Jack symmetric functions. Building on recent results of the second author, we are able to prove new cases of…
Schoenberg showed that a function $f:(-1,1)\rightarrow \mathbb{R}$ such that $C=[c_{ij}]_{i,j}$ positive semi-definite implies that $f(C)=[f(c_{ij})]_{i,j}$ is also positive semi-definite must be analytic and have Taylor series coefficients…
We characterize the $k$-Schur functions as the graded characters of simple objects in an additive module category. This confirms a set of conjectures formulated in the Ph.D. thesis of Chen, written under the direction of Mark Haiman, and…