相关论文: Littlewood-Richardson semigroups
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kra\'skiewicz and developed by Lam and Serrano. We also introduce…
We obtain closed formulas, in terms of Littlewood-Richardson coefficients, for the canonical basis elements of the Fock space representation of $U_v(\hat{\mathfrak{sl}}_e)$ which are labelled by partitions having 'locally small'…
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.…
Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_\lambda$ arise as partition functions of certain path configurations in the $\mathfrak{sl}_2$ higher spin six vertex models. They are multiparameter generalizations of…
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…
The classical Littlewood-Richardson coefficients C(lambda,mu,nu) carry a natural $S_3$ symmetry via permutation of the indices. Our "carton rule" for computing these numbers transparently and uniformly explains these six symmetries;…
We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical…
A new combinatorial approach to the ribbon tableaux generating functions and q-Littlewood Richardson coefficients of Lascoux, Leclerc and Thibon is suggested. We define operators which add ribbons to partitions and following Fomin and…
The group is interesting as the first example of split rank 2 semisimple group, all the irreducible unitary representations of which are known. We make a precise realization of the discrete series representations (in Section 2) by using the…
A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of…
Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in…
For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not…
For $G=GL(n,\mathbb{C})$ and a parabolic subgroup $P=LN$ with a two-block Levi subgroup $L=GL(n_1)\times GL(n_2)$, the space $G\cdot (\mathcal{\mathcal{O}}+\mathfrak{n})$, where $\mathcal{O}$ is a nilpotent orbit of $\mathfrak{l}$, is a…
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 give a closed formula for the number of partitions $\lambda$ of $n$ such that the corresponding irreducible representation $V_\lambda$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
A Littlewood identity is an identity equating a sum of Schur functions with an infinite product. A bounded Littlewood identity is one where the sum is taken over the partitions with a bounded number of rows or columns. The price to pay is…
We consider a type of divided symmetrization $\overrightarrow{D}_{\lambda,G}$ where $\lambda$ is a nonincreasing partition on $n$ and where $G$ is a graph. We discover that in the case where $\lambda$ is a hook shape partition with first…
J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert…