Related papers: Bases for Quotients of Symmetric Polynomials
We investigate chromatic symmetric functions in the relation to the algebra $\Gamma$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $\Gamma$ in terms of chromatic symmetric functions. We also consider…
The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the…
We introduce a family of commuting generalised symmetries of the Dunkl--Dirac operator inspired by the Maxwell construction in harmonic analysis. We use these generalised symmetries to construct bases of the polynomial null-solutions of the…
In this paper, we present a closed formula for the cohomology of real Grassmannians. To achieve this, we use a theory of stratified spaces to compute the differentials in a chain complex that computes the cohomology. Specifically, we…
The eigenfamilies of Gudmundsson and Sakovich can be used to generate harmonic morphisms, proper $r$-harmonic maps, and minimal co-dimension $2$ submanifolds. This article begins by characterising the globally defined eigenfamilies of the…
This is a note for constructing fundamental invariants and computing the Hilbert series of the invariant subalgebras of tensor products of polynomial rings under the action by a direct product of symmetric groups. Our computation relies on…
The main results of this paper establish a partial correspondence between two previously-studied analogues of Groebner bases in the setting of algebras: namely, subalgebra (aka SAGBI) bases for quotients of polynomial rings and Khovanskii…
We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter…
We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with…
We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur…
We show that canonical bases in $\dot{U}(\mathfrak{sl}_n)$ and the Schur algebra are compatible; in fact we extend this result to $p$-canonical bases. This follows immediately from a fullness result from a functor categorifying this map. In…
We define a generalization of the idempotented Schur algebra which gives a unified setting for a quantum group presentation of the Homfly-Pt polynomial, together with its specializations to the Alexander polynomial and the slm…
For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_q^-(\mathfrak{g})$, where $U_q^-(\mathfrak{g})$ is the negative half of the quantized enveloping…
We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation.…
We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful…
The Chern-Schwartz-MacPherson (CSM) and motivic Chern (mC) classes of Schubert cells in a Grassmannian are one parameter deformations of the fundamental classes of the Schubert varieties in cohomology and K-theory respectively. Like the…
In this paper, we give some new and interesting identities which are derived from the basis of Frobenius-Euler. Recently, Simsek et als(see [13]) have given some identities of q-analogue of Frobenius-Euler polynomials related to q-Bernstein…
This article deals with computing the cohomology of Schur functors applied to tautological bundles on super Grassmannians. We show that in a range of cases, the cohomology is a free module over the cohomology of the structure sheaf and that…
The main theorem here is the K-theoretic analogue of the cohomological `stable double component formula' for quiver functions in [Knutson, Miller, and Shimozono, math.AG/0308142]. This K-theoretic version is still in terms of lacing…
We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Pl\"ucker formulas are polynomials in the degree, just as the…