Related papers: Bumpless pipe dreams meet Puzzles

Let P_nk(x) denote the sum of the lowest k+1 terms in the expansion of (1+x)^n. We investigate the irreducibility of P_nk(x) and more general univariate polynomials related to it. Polynomials P_nk(x) naturally arise in Schubert calculus,…

We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by…

We use mirror formulas for the stable quotients analogue of Givental's J-function for twisted projective invariants obtained in a previous paper to obtain mirror formulas for the analogues of the double and triple Givental's J-functions…

The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert…

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,…

Mitosis is a rule introduced by [Knutson-Miller, 2002] for manipulating subsets of the n by n grid. It provides an algorithm that lists the reduced pipe dreams (also known as rc-graphs) [Fomin-Kirillov, Bergeron-Billey] for a permutation w…

Using matrix model, Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau-function as a linear expansion of Schur Q-polynomials. In this paper, we will show directly that the Q-polynomial expansion in this…

We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov-Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we…

Schubert polynomials represent a basis for the cohomology of the complete flag variety and thus play a central role in geometry and combinatorics. In this context, Schubert polynomials are generating functions over various combinatorial…

We investigate the longstanding problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux-M.-P. Sch\"{u}tzenberger (1982) serve…

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram,…

The direct sum map Gr(a,n) x Gr(b,m) -> Gr(a+b,m+n) on Grassmannians induces a K-theory pullback that defines the splitting coefficients. We geometrically explain an identity from [Buch '02] between the splitting coefficients and the…

For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k \binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the investigation of…

A formula for calculating Extensions of (mainly integral) Polynomial Functors is established, based upon projective resolutions. Sample computations are performed, which, in particular, exhibit a surprising non-trivial extension of Divided…

Vakil studied the intersection theory of Schubert varieties in the Grassmannian in a very direct way: he degenerated the intersection of a Schubert variety X_mu and opposite Schubert variety X^nu to a union {X^lambda}, with repetition. This…

This paper gives a tableau formula for expanding the product of a Lascoux polynomial and a stable Grothendieck polynomial into Lascoux polynomials. Lascoux and stable Grothendieck polynomials are inhomogeneous analogues of key polynomials…

In the recent paper [arXiv:1612.06893] P. B\"urgisser and A. Lerario introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $\delta_{k,n}$ the average number of projective $k$-planes in…

Let $\bx_j = \btheta +\bep_j, j=1,...,n$, be observations of an unknown parameter $\btheta$ in a Euclidean or separable Hilbert space $\scrH$, where $\bep_j$ are noises as random elements in $\scrH$ from a general distribution. We study the…

The set of possible spectra (\lambda,\mu,\nu) of zero-sum triples of Hermitian matrices forms a polyhedral cone. We give a complete determination of its facets, finishing a long story with recent highlights by [Helmke-Rosenthal, Klyachko,…

We present the first positive combinatorial rule for expanding the product of a permuted-basement Demazure atom and a Schur polynomial. Special cases of permuted-basement Demazure atoms include Demazure atoms and characters. These cases…