Related papers: Mixed expansion formula for the rectangular Schur …
Building on the work [18], where some standard basis for the queer $q$-Schur superalgebra $\mathcal{Q}_q(n,r;R)$ is defined by a labelling set of matrices and their associated double coset representatives, we investigate the matrix…
It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the…
The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is…
We introduce higher level $q$-oscillator representations for the quantum affine (super)algebras of type $C_n^{(1)},C^{(2)}(n+1)$ and $B^{(1)}(0,n)$. These representations are constructed by applying the fusion procedure to the level one…
The dual immaculate and Young quasisymmetric Schur bases of quasisymmetric functions possess analogues in the peak algebra: respectively, the quasisymmetric Schur $Q$-functions and the peak Young quasisymmetric Schur functions. We show…
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…
The product $s_\mu s_\nu$ of two Schur functions is one of the most famous examples of a Schur-positive function, i.e. a symmetric function which, when written as a linear combination of Schur functions, has all positive coefficients. We…
we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable…
There exist a number of well known multiplicative generating functions for series of Schur functions. Amongst these are some related to the dual Cauchy identity whose expansion coefficients are rather simple, and in some cases periodic in…
Drinfeld's degenerate affine analog of Schur-Weyl duality relates representations of the degenerate affine Hecke algebra $AH_r$ to representations of the Yangian $Y_n$. One way to understand the construction is to introduce an intermediate…
We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline augmented fillings to provide a combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The…
A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear…
We construct a lift of Schur's Q-functions to the peak algebra of the symmetric group, called the noncommutative Schur Q-functions, and extract from them a new natural basis with several nice properties such as the positive right-Pieri…
Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are…
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 introduce the multiple zeta functions with structures similar to those of symmetric functions such as Schur $P$-, Schur $Q$-, symplectic and orthogonal functions in the representation theory. We first consider their basic properties such…
For a function of a type $ \left| \mathbf{r}_1{+}\ldots {+}\mathbf{r}_{_N} \right|^{-\nu} \in \mathbb{R} $ from the many-dimensional vectors $ \mathbf{r}_s $ in Euclidean space, the successive algebraic approach is the derivation of the…
A rational homogeneous (of degree one) positive real matrix-valued function is presented as the Schur complement of a block of the linear pencil with positive semidefinite matrix coefficients. The partial derivative numerators of a rational…
Gorsky and Negut introduced operators $Q_{m,n}$ on symmetric functions and conjectured that, in the case where $m$ and $n$ are relatively prime, the expression ${Q}_{m,n}(1)$ is given by the Hikita polynomial ${H}_{m,n}[X;q,t]$. Later,…
Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and…