The classification of multiplicity-free plethysms of Schur functions

The uniqueness of plethystic factorisation

We prove that a plethysm product of two Schur functions can be factorised uniquely and classify homogeneous and indecomposable plethysm products.

Schur函子与范畴化Plethysm

Schur Functors and Categorified Plethysm

本文证明Schur函子范畴构成一个“2-plethory”，其下降诱导了对称函数环的plethory结构，从而揭示对称函数的丰富结构在更高范畴层次上的存在。

Splitting the square of homogeneous and elementary functions into their symmetric and anti-symmetric parts

We consider the expansion of the square of a complete homogeneous function $h_λ$, or of an elementary symmetric function $e_λ$, in the basis of Schur functions. This square also decomposes into two plethysms, $s_2[h_λ]$ and $s_{11}[h_λ]$ (resp. $s_2[h_λ]$ and $s_{11}[h_λ]$), which are called its symmetric and anti-symmetric parts, respectively. We define a sign statistic on the set of tableaux that index the Schur functions appearing in the square of those symmetric functions. This sign statistic allows to determine to which plethysm each Schur function contributes. We use mainly combinatorial…

Stabilized plethysms for the classical Lie groups

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finit…

A generalized SXP rule proved by bijections and involutions

This paper proves a combinatorial rule expressing the product $s_τ(s_{λ/μ} \circ p_r)$ of a Schur function and the plethysm of a skew Schur function with a power sum symmetric function as an integral linear combination of Schur functions. This generalizes the SXP rule for the plethysm $s_λ\circ p_r$. Each step in the proof uses either an explicit bijection or a sign-reversing involution. The proof is inspired by an earlier proof of the SXP rule due to Remmel and Shimozono, A simple proof of the Littlewood--Richardson rule and applications, Discrete Mathematics 193 (1998) 257--266. The connecti…

Plethysm and the algebra of uniform block permutations

We study the representation theory of the uniform block permutation algebra in the context of the representation theory of factorizable inverse monoids. The uniform block permutation algebra is a subalgebra of the partition algebra and is also known as the party algebra. We compute its characters and provide a Frobenius characteristic map to symmetric functions. This reveals connections of the characters of the uniform block permutation algebra and plethysms of Schur functions.

Rectangular Schur functions and the basic representation of affine Lie algebras

An expression is given for the plethysm $p_{2}\circ S_{\square}$, where $p_{2}$ is the power sum of degree two and $S_{\square}$ is the Schur function indexed by a rectangular partition. The formula can be well understood from the viewpoint of the basic representation of the affine Lie algebra of type $A_{2}^{(2)}$.

Plethysms of symmetric functions and representations of $\mathrm{SL}_2(\mathbb{C})$

Let $\nabla^λ$ denote the Schur functor labelled by the partition $λ$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^λ\!\mathrm{Sym}^\ell \!E \cong \nabla^μ\!\mathrm{Sym}^m \! E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $λ$ and $μ$ are conjugate partitions and when one of $λ$ or $μ$ is a rectangle. We give a complete classification when $λ$ and $μ$ each have at most two rows or columns or is a hook parti…

Plethysms of symmetric functions and highest weight representations

Let $s_ν\circ s_μ$ denote the plethystic product of the Schur functions $s_ν$ and $s_μ$. In this article we define an explicit polynomial representation corresponding to $s_ν\circ s_μ$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\langle s_ν\circ s_μ, s_λ\rangle$ to be stable under insertion of new parts into $μ$ and $λ$. We also characterize all maximal and mini…

plethysm $s_λ[s_m]$ 中不可约表示的数量

The number of irreducibles in the plethysm $s_λ[s_m]$

本文给出了 Schur 函数 plethysm 分解中不可约表示数量的格点计数公式，并在单分量情形下给出组合解释，进而提出 Foulkes 猜想的较弱版本。

特征分解、$z$-非对称分拆与plethysm

Character factorisations, $z$-asymmetric partitions and plethysm

本文利用Hamel和King定义的一般对称函数，将辛群与正交群的特征标分解嵌入到涉及整数$z$和参数$q$的无穷族中，并基于$z$-非对称分拆的新刻画推广了已知分类。

Multiplicity-free skew Schur polynomials

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.

Multiplicity-free Skew Schur functions with full interval support

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.

Plethysm 系数的稳定性与修正多项式归纳

Stability of plethysm coefficients and modified polynomial induction

本文给出 plethysm 系数两个稳定性结果的简短证明，并建立稳定值与 Weyl 模重数的新联系，提出稳定 Foulkes 系数的向量划分函数公式。

Haglund's positivity conjecture for multiplicity one pairs

Haglund's conjecture states that $\dfrac{\langle J_λ(q,q^k),s_μ\rangle}{(1-q)^{|λ|}} \in \mathbb{Z}_{\geq 0}[q]$ for all partitions $λ,μ$ and all non-negative integers $k$, where $J_λ$ is the integral form Macdonald symmetric function and $s_μ$ is the Schur function. This paper proves Haglund's conjecture in the cases when the pair $(λ,μ)$ satisfies $K_{λ,μ}=1$ or $K_{μ',λ'}=1$ where $K$ denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.

The Kronecker product of Schur functions indexed by two-row shapes or hook shapes

The Kronecker product of two Schur functions $s_μ$ and $s_ν$, denoted by $s_μ*s_ν$, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions $μ$ and $ν$. The coefficient of $s_λ$ in this product is denoted by $γ^λ_{μν}$, and corresponds to the multiplicity of the irreducible character $χ^λ$ in $χ^μχ^ν.$ We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for $s_λ[XY]$ to find closed formulas for the Kronecker coefficients $γ^λ_{μν}$ when $λ$ is an…

A_N Multiplicity Rules And Schur Functions

We show that a specialization in Weyl character formula can be carried out in such a way that its right-hand side becomes simply a Schur Function. For this, we need the use of fundamental weights. In the generic definition, an Elementary Schur Function $S_Q(x_1,x_2,..,x_Q)$ of degree Q is known to be defined by some polynomial of Q indeterminates $ x_1,x_2,..,x_Q $. It is also known that definition of Elementary Schur Functions can be generalized in such a way that for any partition $(Q_k)$ of weight Q and length k one has a Generalized Schur Function $S_{(Q_k)}(x_1,x_2,..,x_Q)$. When they are…

Spin invariant theory for the symmetric group

We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur Q-functions and a shifted q-hook formula. In addition, we provide a bijective proof for a formula of the principal specialization of the Schur Q-functions.

Module varieties and representation type of finite-dimensional algebras

In this paper we seek geometric and invariant-theoretic characterizations of (Schur-)representation finite algebras. To this end, we introduce two classes of finite-dimensional algebras: those with the dense-orbit property and those with the multiplicity-free property. We show first that when a connected algebra A admits a preprojective component, each of these properties is equivalent to A being representation-finite. Next, we give an example of an algebra which is not representation-finite but still has the dense-orbit property. We also show that the string algebras with the dense orbit-prop…

有限约化群的特征标重数与Drinfeld对偶

Multiplicity of characters of finite reductive groups and Drinfeld doubles

研究有限约化群\(G^F\)的几乎幂幺特征标和德利涅 - 卢茨蒂格特征标张量积重数，涉及与复不可约特征标环结构关系，还考虑Drinfeld对偶上模的弗罗贝尼乌斯 - 舒尔指标，最后研究相关重数并提出检测问题。

克罗内克系数与同时共轭类

Kronecker Coefficients and Simultaneous Conjugacy Classes

该研究引入克罗内克赫克代数研究有限群的克罗内克系数，建立同时共轭类数量与克罗内克系数的关系，定义双实群并刻画其与无重数张量积的关联，证明多类有限群不具有无重数张量积。

Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients and Regular Functions

In earlier papers we studied direct limits $(G,K) = \varinjlim (G_n,K_n)$ of two types of Gelfand pairs. The first type was that in which the $G_n/K_n$ are compact Riemannian symmetric spaces. The second type was that in which $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius--Schur Orthogonality Relations to define isometric injections $ζ_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)$ for $m \geqq n$ and prove that the left regular representation of $G$ on …