Richard P. Stanley
We give a new formula for the values of an irreducible character of the symmetric group S_n indexed by a partition of rectangular shape. Some observations and a conjecture are given concerning a generalization to arbitrary shapes.
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We consider the decomposition into irreducible components of the external power $Λ^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(λ,μ)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$μ=λ'$. We show that the Young diagram $λ$ which corresponds to a randomly selected irreducible component $(λ,λ')$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a random Young tableau of rectangular …
We derive several identities that feature irreducible characters of the general linear, the symplectic, the orthogonal, and the special orthogonal groups. All the identities feature characters that are indexed by shapes that are "nearly" rectangular, by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others. As applications we prove new results in plane partitions and tableaux enumeration, including new refinements of the Bender-Knuth and MacMahon (ex-)conjectures.
In math.CO/0109093 the author obtained a formula for the value of an irreducible symmetric group character indexed by a partition of rectangular shape. In the present paper this formula is (conjecturally) generalized to arbitrary shapes.
R. Stanley has found a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, he has given a conjectural generalisation for any shape. Here, we will prove this formula using shifted Schur functions and Jucys-Murphy elements.
We give closed product formulas for the irreducible characters of the symmetric groups related to rectangular `almost square' Young diagrams $p \times(p+δ)$ for a fixed value of an integer $δ$ and an arbitrary integer $p$.
We show that certain weighted average of the alpha-determinant of a $kn$ by $kn$ matrix of the form $A\otimes1_{1,k}$, the Kronecker product of a $kn$ by $n$ matrix $A$ and $1$ by $k$ all one matrix $1_{1,k}$, over permutations of $kn$ letters is reduced to the $k$-wreath determinant of $A$ up to constant. The constant is exactly given by the modified content polynomial for the Young diagram $(k^n)$. As a corollary, we give a `determinantal' formula for certain functions on the symmetric groups which are invariant under the left and right translation by a Young subgroup, especially the values …
The complex irreducible representations of the symmetric group carry an important canonical basis called the Kazhdan-Lusztig basis. Although it is difficult to express how general permutations act on this basis, some distinguished permutations have beautiful descriptions. In 2010 Rhoades showed that the long cycle $(1, 2,..., n)$ acts by the jeu-de-taquin promotion operator in the case when the irreducible representation is indexed by a rectangular partition. We prove a generalisation of this theorem in two directions: on the one hand we lift the restriction on the shape of the partition, and …
We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if $M$ is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus tilings of a hexagon, which has an equivalent formulation in terms of plane partitions. A simila…
For any finite group G it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in a given characteristic p. We answer this question for p=2 when G is the proper double cover of the alternating group. As a key ingredient in the proof, we prove a formula for the decomposition numbers in Rouquier blocks of double covers of symmetric groups, in terms of Schur P-functions.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
We consider sequences of degrees of ordinary irreducible $S_n$-characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of $n$ with leading coefficient less than one. We show that any such sequence has at least exponential growth and we compute an explicit bound.
Combinatorics of irreducible characters for Lie superalgebra $\frak{gl}(m,n)$
本文利用 Su-Zhang 公式与 Brion 定理，为有限维不可约 $\ rak{gl}(m,n)$ 模的特征标提供了新公式。
Suppose that $χ_λ$ and $χ_μ$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $π\in S_n$ such that $χ_λ(π)$ is provably different from $χ_μ(π)$. In fact, we show a little more. Suppose $f=χ_λ$ for some irreducible character $χ_λ$ of $S_n$, but we do not know $λ$, and we are given only oracle access to $f$. We give an algorithm that determines $λ$, using a number of queries to $f$ that is polynomial in $n$. Each query can be computed in time polynomial in $n$ by someone who knows $λ$.
Bijection between trees in Stanley character formula and factorizations of a cycle
本文针对 Stanley 特征多项式最高次项系数对应的带装饰平面树，构造了其与轮换最小因子分解之间的明确双射。
Ramanujan sums and rectangular power sums
该研究探讨了由拉马努金和与幂和对称函数构造的对称函数，证明了u=0和u=1时的Schur正性并给出对称群表示的分解，还猜想u=2时同样具有Schur正性。
An irreducible character $χ$ of an association scheme is called nonlinear if the multiplicity of $χ$ is greater than $1$. The main result of this paper gives a characterization of commutative association schemes with at most two nonlinear irreducible characters. This yields a characterization of finite groups with at most two nonlinear irreducible characters. A class of noncommutative association schemes with at most two nonlinear irreducible character is also given.
We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Ram's combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrac…
The aim of this work is to extend to a general $S_m\times S_n$-module context the Grossman-Bizley paradigm that allows the enumeration of Dyck paths in a $m\times n$-rectangle. We obtain an explicit formula for the the "bi-Frobenius" characteristic of what we call {\em interlaced} rectangular parking functions in an $m\times n$-rectangle. These are obtained by labelling the $n$ vertical steps of an $m\times n$-Dyck path by the numbers from $1$ to $n$, together with an independent labelling of its horizontal steps by integers from $1$ to $m$. Our formula specializes to give the Frobenius charac…
We construct a subalgebra of dimension $2.3^{n-1}$ of the group algebra of a Weyl group of type $B_n$ containing its Solomon descent's algebra but also the Solomon's descent algebra of the symmetric group. This lead us to a construction of the irreducible characters of the hyperoctahedral groups by using a generalized plactic equivalence.
The object of this short note is to prove a theorem and present a conjecture for the number of even entries in the character table of the symmetric group.
Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers dividing some character degree and of the non-identity character degrees, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. This graph is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this pap…
A recent approach to the Beck-Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoss (SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main tool is a conditional second moment method combined with Stein's method of exch…
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