Related papers: Large tensor products and Littlewood-Richardson co…
This paper focuses on the $GL_n$ tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary finite dimensional irreducible representations of $GL_n$. We will describe an explicit basis for this algebra.…
Let G be a semisimple algebraic group over an algebraically-closed field of characteristic zero. In this note we show that every regular face of the Littlewood-Richardson cone of G gives rise to a reduction rule: a rule which, given a…
A reduction formula for the branching coefficients of tensor products of representations and more generally restrictions of representations of a semisimple group to a semisimple subgroup is proved in work by Knutson-Tao and Derksen-Weyman.…
The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of GL(n). Using the hive model for the…
We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there…
We determine the graded decompositions of fusion products of finite-dimensional irreducible representations for simple Lie algebras of rank two. Moreover, we give generators and relations for these representations and obtain as a…
The Littlewood-Richardson (LR) coefficient counts among many other things the LR tableaux of a given shape and a given content. We prove, that the number of LR tableaux weakly increases if one adds to the shape and the content the shape and…
Through a reformulation of the local limit theorem and law of small numbers, which is obtained by working in the spaces naturally associated to the limiting distributions, we discover a general and abstract framework for the investigation…
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…
We provide direct proofs of product and coproduct formulae for Schur functions where the coefficients (Littlewood--Richardson coefficients) are defined as counting puzzles. The product formula includes a second alphabet for the Schur…
It is known that the Littlewood-Richardson coefficients can be calculated using a certain class of measures, and these measures have a rigidity property when the coefficient is equal to 1. Rigid measures decompose uniquely into sums of…
We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are…
The Littlewood-Richardson coefficients $c^\nu_{\lambda,\mu}$ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL$(n, {\mathbb C})$. They are parametrized by the…
The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K.…
We present a program that allows for the computation of tensor products of irreducible representations of Lie algebras A-G based on the explicit construction of weight states. This straightforward approach (which is slower and more…
Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups ($GL_n$). The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for…
The basic representation of $\A$ is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.
This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute ``Symmetric functions 2001: surveys of developments and perspectives" (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June…
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 study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random…