Related papers: Powers of the Vandermonde determinant, Schur Funct…
We present a single operation for constructing skew diagrams whose corresponding skew Schur functions are equal. This combinatorial operation naturally generalises and unifies all results of this type to date. Moreover, our operation…
Littlewood-Richardson rule gives the decomposition formula for the multiplication of two Schur functions, while the decomposition formula for the multiplication of two Hall-Littlewood functions or two universal characters is also given by…
The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…
An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of…
We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown…
Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we…
A decomposition of any symmetric power of $\Bbb C^2\otimes\Bbb C^2\otimes\Bbb C^2$ into irreducible $sl_2(\Bbb C)\oplus sl_2(\Bbb C)\oplus sl_2(\Bbb C)$-submodules are presented. Namely, the multiplicities of irreducible summands in the…
Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current…
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…
The present paper is a detailed version of math/0003031. We introduce and study a new basis in the algebra of symmetric functions. The elements of this basis are called the Frobenius-Schur functions (FS-functions, for short). Our main…
In this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields $2^n$ distinct and unique decompositions for any slice function with domain in $\mathbb{H}^n$. Depending…
In this paper we give a symbolical formula and a cancellation-free formula for the Schur elements associated to the simple modules of the degenerate cyclotomic Hecke algebras. As some direct applications, we show that the Schur elements are…
We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups S_n contain all irreducibles as their constituents. Our main result is that they contain representations…
We show that the left and right adjoint of the Schur functor can be expressed in terms of the monoidal structure of strict polynomial functors. Using this result we give a necessary and sufficient condition for when the tensor product of…
We prove a conjecture of Bourn and Willenbring (2020) regarding the palindromicity and unimodality of a certain family of polynomials $N_n(t)$. These recursively defined polynomials arise as the numerators of generating functions in the…
Given a topological modular functor $\mathcal{V}$ in the sense of Walker \cite{Walker}, we construct vector bundles over $\bar{\mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction…
Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of "mixed" products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine…
In this tutorial, exponentiation and factorization (decomposition) formulas are derived and discussed for common matrix operators that arise in studies of classical dynamics, linear and nonlinear optics, and special relativity. To…
We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight \[w(x;t)=|x|^{2\lambda+1}\exp\left(-x^4+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $\lambda>-1$…
Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable as Schur-Szeg\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give…