相关论文: Families of major index distributions: closed form…

We introduce a Pfaffian formula that extends Schur's $Q$-functions $Q_\lambda$ to be indexed by compositions $\lambda$ with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the…

Let $\mathrm{JT}_\lambda$ be the Jacobi-Trudi matrix corresponding to the partition $\lambda$, so $\det\mathrm{JT}_\lambda$ is the Schur function $s_\lambda$ in the variables $x_1,x_2,\dots$. Set $x_1=\cdots=x_n=1$ and all other $x_i=0$.…

We investigate the unflavoured Schur indices of class $\mathcal S$ theories of modest rank, and in the case of $\mathcal{N}=4$ super Yang-Mills theory with special unitary gauge group of somewhat more general rank, with an eye towards their…

By considering the specialisation $s_{\lambda}(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $\lambda$ in terms of two properties of the boxes in…

We introduce a new family of Schur functions $s_{\lambda/\mu;a,b}(x/y)$ that depend on two sets of variables and two sequences of parameters. These free fermionic Schur functions have a hidden symmetry between the two sets of parameters…

We find closed-form expressions for the Schur indices of 4d $\mathcal{N}=2^{*}$ super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams…

We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a…

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 consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We…

We attempt to explain the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations…

We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We use cumulants to classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary…

We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $\omega$ on the ring of…

Let $k$ be a commutative $\mathbb{Q}$-algebra. We study families of functors between categories of finitely generated $R$-modules which are defined for all commutative $k$-algebras $R$ simultaneously and are compatible with base changes.…

We discuss several well known results about Schur functions that can be proved using cancellations in alternating summations; notably we shall discuss the Pieri and Murnaghan-Nakayama rules, the Jacobi-Trudi identity and its dual (Von…

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized…

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur…

We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{\mu\le\lambda} g^{(k)}_{\mu}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by…

Recently the Schur index of ${\cal N}=4$ SYM was evaluated in closed form to all orders including exponential corrections in the large $N$ expansion and for fixed finite $N$. This was achieved by identifying the matrix model which…

Macdonald's ninth variation of Schur functions is a broad generalization of the classical Schur function and its variants, defined via the Jacobi-Trudi determinant formula. In this paper, we establish various algebraic relations for…

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our…