分拆代数塔上胞模的分支法则

An Elliptic Hypergeometric Function Approach to Branching Rules

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures exhibiting a novel type of vanishing behaviour involving partitions with empty 2-cores.

Representations of symmetric groups with non-trivial determinant

We give a closed formula for the number of partitions $λ$ of $n$ such that the corresponding irreducible representation $V_λ$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant.

带电立体分拆

Charging solid partitions

本文研究4维熔融晶体对应的立体分拆，推导其电荷函数，并发现4盒与5盒簇贡献导致相应BPS代数构造更复杂。

Tonal partition algebras: fundamental and geometrical aspects of representation theory

For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[δ]$. We construct modules $\{ Δ_{\underlineμ} \}_{\underlineμ}$ for $P^l_n$ over $\mathbb{Z}[δ]$, and hence over any integral domain containing $\mathbb{Z}[δ]$ that is a $\mathbb{Z}[δ]$-algebra (such as $\mathbb{C}[δ]$), that pass to a complete set of irreducible modules over the field of fractions. We show that $P^l_n$ is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer [6]. (The aim is to investigate the non-semisimple structure of the tonal part…

Branching Rules for Specht Modules

Let n be a positive integer and let Sigma_n be the symmetric group of degree n. Let S^lambda be the Specht module for Sigma_n corresponding to a partition lambda of n, defined over a field F of odd characteristic. We find the indecomposable components of the restriction of S^lambda to Sigma_{n-1}, and of the induction of S^lambda to Sigma_{n+1}. Namely, if b and B are block idempotents of FSigma_{n-1} and FSigma_{n+1} respectively, then the modules S^lambda b and S^lambda B are 0 or indecomposable. We give examples to show that the assumption that F has odd characteristic cannot be dropped.

Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals

Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in \cite{Chi08}, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of $\gl(n)$ as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood-Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its…

Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras

We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size $N$. We solve the general 11- parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras $P_k(N)$. The integer $k$ is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Fam…

Carter-Payne homomorphisms and branching rules for endomorphism rings of Specht modules

Let n be a positive integer and let p be a prime. Suppose that we take a partition of n, and obtain another partition by moving a node from one row to a shorther row. Carter and Payne showed that if the p-residue of the removed and added positions is the same, then there is a non-zero homomorphism between the corresponding Specht modules for the symmetric group of degree n, defined over a field of characteristic p. In this paper we give a very simple description of such a homomorphism, as a map between polytabloids, using the action of a Murphy-Jucys element. We also present a proof that in th…

Induction and restriction functors for cellular categories

Cellular categories are a generalization of cellular algebras, which include a number of important categories such as (affine)Temperley-Lieb categories, Brauer diagram categories, partition categories, the categories of invariant tensors for certain quantised enveloping algebras and their highest weight representations, Hecke categories and so on. The common feather is that, for most of the examples, the endomorphism algebras of the categories form a tower of algebras. In this paper, we give an axiomatic framework for the cellular categories related to the quasi-hereditary tower and then study…

Traces On Diagram Algebras I: Free Partition Quantum Groups, Random Lattice Paths And Random Walks On Trees

We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica-Speicher (arXiv:0808.2628) and Weber (arXiv:1201.4723). For the infinite-dimensional Temperley-Lieb-algebra (corresponding to the quantum group $O^+_N$) and the Motzkin algebra ($B^+_N$), the classification of extremal traces implies a classification result for well-known types of central random lattice paths. For the $2$-Fuss-Catalan algebra ($H_N^+$) we solve the classification problem by computing the \emph{minimal or exit bounda…

The Spherical Hecke algebra, partition functions, and motivic integration

This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition fu…

Elementary divisors of Gram matrices of certain Specht modules

The elementary divisors of the Gram matrices of Specht modules S^lambda over the symmetric group are determined for two-row partitions and for two-column partitions lambda. More precisely, the subquotients of the Jantzen filtration are calculated using Schaper's formula. Moreover, considering a general partition lambda of n at a prime p > n - lambda_1, the only possible non trivial composition factor of S_p^lambda is induced by the morphism of Carter and Payne, as shown by means of Kleshchev's modular branching rule. This enables the Jantzen filtration to be calculated in this case as well.

GL_N(F)的所有零深度表示的Zelevinsky段与超特殊分支律

Zelevinsky Segments and Hyperspecial Branching Laws for All Depth-Zero Representations of GL_N(F)

该研究针对非阿基米德局部域上GL_N(F)的不可约零深度表示，结合Bushnell-Kutzko型等工具，确定了超特殊极大紧子群幂幺根基固定向量空间的分解，建立了p进与有限侧对偶的交织关系，得到了完整有效的超特殊分支律。

Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$

The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q)\subseteq U_n(q)$ for $m\leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the superch…

The complexity of certain Specht modules for the symmetric group

During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules S^λfor the symmetric group, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The Georgia group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of $λ$ is built out of $p \times p$ blocks. We prove one direction of this conjecture by showing these Specht modu…