Christian Gutschwager
In this paper we show that two skew diagrams lambda/mu and alpha/beta can represent the same multiplicity free skew character [lambda/mu]=[alpha/beta] only in the the trivial cases when lambda/mu and alpha/beta are the same up to translation or rotation or if lambda=alpha is a staircase partition lambda=(l,l-1,...,2,1) and lambda/mu and alpha/beta are conjugate of each other.
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We provide a non-recursive, combinatorial classification of multiplicity-free skew Schur polynomials. These polynomials are $GL_n$, and $SL_n$, characters of the skew Schur modules. Our result extends work of H. Thomas--A. Yong, and C. Gutschwager, in which they classify the multiplicity-free skew Schur functions.
It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e. skew Schur functions having Littlewood-Richardson coefficients always equal to $1$ over the full interval.
We find a simple criterion for the equality $Q_λ=Q_{μ/ν}$ where $Q_λ$ and $Q_{μ/ν}$ are Schur's Q-functions on infinitely many variables.
We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables.
We classify the $Q$-multiplicity-free skew Schur $Q$-functions. Towards this result, we also provide new relations between the shifted Littlewood-Richardson coefficients.
The Schur polynomials $s_λ$ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For $ρ= (n, n-1, \dots, 1)$ a staircase shape and $μ\subseteq ρ$ a subpartition, the Stembridge equality states that $s_{ρ/μ} = s_{ρ/μ^T}$. This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials $G_λ$, and the dual stable Grothendieck polynomials $g_λ$, developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the $K$-theory of the…
In this paper we classify the multiplicity-free skew characters of the symmetric group. Furthermore we show that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the `inverse' partitions (Theorem 4.2).
We introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover, we determine all relations between ribbon Schur Q-functions; show they supply a Z-basis for skew Schur Q-functions; assert their irreducibility; and show that the non-commutative analogue of ribbon Schur Q-functions is the…
In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape…
The hive model is a combinatorial device that may be used to determine Littlewood-Richardson coefficients and study their properties. It represents an alternative to the use of the Littlewood-Richardson rule. Here properties of hives are used to determine all possible multiplicity-free Schur function products and skew Schur function expansions. This confirms the results of Stembridge, Gutschwager, and Thomas and Yong, and sheds light on the combinatorial origin of the conditions for being multiplicity-free, as well as illustrating some of the key features and power of the hive model.
We provide a classification of multiplicity-free inner tensor products of irreducible characters of symmetric groups, thus confirming a conjecture of Bessenrodt. Concurrently, we classify all multiplicity-free inner tensor products of skew characters of the symmetric groups. We also provide formulae for calculating the decomposition of these tensor products.
Equality of skew Schur functions in noncommuting variables
本文解决了非交换变量中两个斜 Schur 函数何时相等的分类问题，给出了当斜图不同时二者相等当且仅当一个为非对称 ribbon、另一个为其对径旋转且置换满足显式双射条件的完整刻画。
In this paper, we interpret the multiplicity of 1 in Laplacian spectra of trees and prove that Faria's inequality turns to an equality in the case of normal trees which yields that in any tree without a vertex of degree 2, Faria equality holds and multiplicity of 1 in Laplacian spectrum will be equal to star degree of the the tree. As a result we introduce a combinatorial procedure for computing the multiplicity of 1. In the way to prove this results we will introduce many transformation on graphs which are invariant regarding to the multiplicity of 1. We also introduce an inequality for the m…
To every labeled poset (P,ω), one can associate a quasisymmetric generating function for its (P,ω)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P,ω) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of line…
A finite set $X$ in a complex sphere is called a complex spherical $2$-code if the number of inner products between two distinct vectors in $X$ is equal to $2$. In this paper, we characterize the tight complex spherical $2$-codes by doubly regular tournaments, or skew Hadamard matrices. We also give certain maximal 2-codes relating to skew-symmetric $D$-optimal designs. To prove them, we show the smallest embedding dimension of a tournament into a complex sphere by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel matrix.
It is known that the Kronecker coefficient of three partitions is a bounded and weakly increasing sequence if one increases the first part of all three partitions. Furthermore if the first parts of partitions λ,μare big enough then the coefficients of the Kronecker product [λ][μ]=\sum_\n g(ł,\m,\n)[ν] do not depend on the first part but only on the other parts. The reduced Kronecker product [λ]_\bullet \star[μ]_\bullet can be viewed (roughly) as the Kronecker product [(n-|λ|,λ)][(n-|μ|,\m)] for n big enough. In this paper we classify the reduced Kronecker products which are multiplicity free a…
Skew Howe duality and limit shapes of Young diagrams
本文利用晶体基建立斜Howe对偶的组合版本，将表示重数表示为行列式与乘积公式，并证明其q-类似等于q-维数，进而取无穷秩极限证明Young图一致收敛到极限形状。
There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that…
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