相关论文: Pattern bounds for principal specializations of $\…

Motivated by Stanley's ``Schubert shenanigans'' paper, commendable attempts have been made to understand the principal specializations of Schubert or Grothendieck polynomials. In this paper, we prove that when a permutation $w$ does not…

We show that the principal specialization of the Schubert polynomial at $w$ is bounded below by $1+p_{132}(w)+p_{1432}(w)$ where $p_u(w)$ is the number of occurrences of the pattern $u$ in $w$, strengthening a previous result by A.…

This paper investigates the number of supports of the Schubert polynomial $\mathfrak{S}_w(x)$ indexed by a permutation $w$. This number also equals the number of lattice points in the Newton polytope of $\mathfrak{S}_w(x)$. We establish a…

Let $v(n)$ be the largest principal specialization of Schubert polynomials for layered permutations $v(n) := \max_{w \in \mathcal{L}_n} \mathfrak{S}_w(1,\ldots,1)$. Morales, Pak and Panova proved that there is a limit \[\lim_{n \to \infty}…

Let $w$ be a permutation of $\{1,2,\ldots,n \}$, and let $D(w)$ be the Rothe diagram of $w$. The Schubert polynomial $\mathfrak{S}_w(x)$ can be realized as the dual character of the flagged Weyl module associated to $D(w)$. This implies a…

Denote by $u(n)$ the largest principal specialization of the Schubert polynomial: $ u(n) := \max_{w \in S_n} \mathfrak{S}_w(1,\ldots,1) $ Stanley conjectured in [arXiv:1704.00851] that there is a limit $\lim_{n\to \infty} \, \frac{1}{n^2}…

We find a layered permutation $w\in S_n$ whose Schubert polynomial $\mathfrak S_w(x_1, \dots, x_n)$ has support of size asymptotically at least $n!/4^n$. This gives precise asymptotics for the growth rate of $\beta(n):= \max_{w\in…

We present computational results on principal specializations $\mathfrak{S}_w(1^n)$ of Schubert polynomials, which count reduced pipe dreams and reduced bumpless pipe dreams (RBPD). We find the first counterexample, at $n=17$, to the…

We prove a sharp lower bound on the number of terms in an element of the reduced Gr\"obner basis of a Schubert determinantal ideal $I_w$ under the term order of [Knutson-Miller '05]. We give three applications. First, we give a…

Schubert polynomials $\mathfrak{S}_w$ are polynomial representatives for cohomology classes of Schubert varieties in a complete flag variety, while Grothendieck polynomials $\mathfrak{G}_w$ are analogous representatives for the $K$-theory…

We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about…

There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently…

We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $\beta=1$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional…

Grothendieck polynomials $\mathfrak{G}_w$ of permutations $w\in S_n$ were introduced by Lascoux and Sch\"utzenberger in 1982 as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the…

Fink, M\'esz\'aros and St.Dizier showed that the Schubert polynomial $\mathfrak{S}_w(x)$ is zero-one if and only if $w$ avoids twelve permutation patterns. In this paper, we prove that the Grothendieck polynomial $\mathfrak{G}_w(x)$ is…

We show that for any permutation $w$ that avoids a certain set of 13 patterns of lengths 5 and 6, the Schubert polynomial $\mathfrak S_w$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner…

We prove a formula for the degrees of Ikeda and Naruse's $P$-Grothendieck polynomials using combinatorics of shifted tableaux. We show this formula can be used in conjunction with results of Hamaker, Marberg, and Pawlowski to obtain an…

We give a combinatorial formula for the Kazhdan-Lusztig polynomials $P_{x,w}$ in the symmetric group when $w$ is a 321-hexagon-avoiding permutation. Our formula, which depends on a combinatorial framework developed by Deodhar, can be…

We prove that if $\sigma \in S_m$ is a pattern of $w \in S_n$, then we can express the Schubert polynomial $\mathfrak{S}_w$ as a monomial times $\mathfrak{S}_\sigma$ (in reindexed variables) plus a polynomial with nonnegative coefficients.…

We give an algebra-combinatorial constructions of (noncommutative) generating functions of double Schubert and double $\beta$-Grothendieck polynomials corresponding to the full flag varieties associated to the Lie groups of classical types…