Related papers: An Introduction to Schur Polynomials
We consider the problem of embedding the semi-ring of Schur-positive symmetric polynomials into its analogue for the classical types $B/C/D$. If we preserve highest weights and add the additional Lie-theoretic parity assumption that the…
In this contribution we deal with Gaussian quadrature rules based on orthogonal polynomials associated with a weight function $w(x)= x^{\alpha} e^{-x}$ supported on an interval $(0,z)$, $z>0.$ The modified Chebyshev algorithm is used in…
We use Young's raising operators to give short and uniform proofs of several well known results about Schur polynomials and symmetric functions, starting from the Jacobi-Trudi identity.
We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_\lambda$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive…
We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as…
We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In…
Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the…
We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is…
The Schur multiple zeta function was defined as a multivariable function by Nakasuji-Phuksuwan-Yamasaki. Inspired by the product formula of Schur functions, the products of Schur multiple zeta functions have been studied. While the product…
We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we…
Grassmann (or anti-commuting) variables are extensively used in theoretical physics. In this paper we use Grassmann variable calculus to give new proofs of celebrated combinatorial identities such as the Lindstr\"om-Gessel-Viennot formula…
The expansion of a Schubert polynomial into slide polynomials corresponds to a sum over sub-balls in the subword complex. There has been recent interest in other, coarser, expansions of Schubert polynomials. We extend the methods used in…
It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise…
In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. These polynomials represent the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by…
We consider the expansion of the square of a complete homogeneous function $h_\lambda$, or of an elementary symmetric function $e_\lambda$, in the basis of Schur functions. This square also decomposes into two plethysms, $s_2[h_\lambda]$…
This is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya,…
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing…
We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of…
We examine and present new combinatorics for the Schur polynomials from the viewpoint of quantum integrability. We introduce and analyze an integrable six-vertex model which can be viewed as a certain degeneration model from a t-deformed…
This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed…