77 problems

Huh–Matherne–Mészáros–St. Dizier conjecture. The support of a homogenized Grothendieck polynomial is precisely the set of integer points of some generalized permutahedron.

Gao's conjecture. For any permutation $w$, $c_w\geq 0$. Equivalently,

Grothendieck polynomials $\mathfrak G_w$ are multivariate polynomials associated to permutations $w\in S_n$. Their homogenization is obtained by introducing an additional homogeniz…

Let $\mathfrak{G}_w$ be a Grothendieck polynomial and let $\widehat{\mathfrak{G}}_w$ denote its top-degree component. Mészáros–Setiabrata–St Dizier's support conjecture. The suppor…

Let $\mu_{m,n}$ be the Plancherel-Hecke probability measure on partitions $\lambda\subset\delta_n=(n,n-1,\ldots,1)$, defined by … where $d^{\lambda}(n)$ counts increasing tableaux…

Let $S_n$ be the symmetric group, let ${\cal G}_w(x_1,\ldots,x_n)$ be the Grothendieck polynomial associated with $w\in S_n$, and let $G_i^{A_{n-1}}$ be the elements defined by…

For a permutation $w$, let $\mathfrak{S}_w(x)$ be the single Schubert polynomial and let $\mathfrak{G}_w(x)$ be the single Grothendieck polynomial. Lascoux's positivity conjecture.…

Let $G_\lambda$ denote the stable polynomial associated with a partition $\lambda$, and let $G_R$ denote the element associated with a rectangular partition $R$. The elements corre…

Let $u$ be a permutation. An affine Grothendieck--Lascoux expansion conjecture asserts that there exist positive integers $c_1,\dotsc,c_t$, nonnegative integers $e_1,\dotsc,e_t$, a…

Let $u,v\in S_n$ be vexillary permutations, let $\phi(w)$ be the Wachs flag of $w$, and write $a\preceq b$ when $a$ is a prefix of $b$. Let $\mathfrak G_w^{(\beta)}(\mathbf x)$ be…

Let $\mathcal P$ denote the set of partitions, let $\widetilde{G}_{\lambda}$ be the modified stable Grothendieck symmetric function, and let $\leqslant_{\operatorname{g}}$ denote G…

Robichaux's ghost Kohnert conjecture. For every permutation $w\in S_n$ and $\alpha\in\mathbb{Z}_{\geq 0}^n$,

Ross–Yong's Grothendieck-polynomial conjecture. For every permutation $w\in S_n$,

Robichaux's support conjecture. Robichaux's rule correctly computes the support of Grothendieck polynomials: for every permutation $w\in S_n$,

Ross–Yong's conjecture. Their rule provided a combinatorial model for both Grothendieck polynomials and Lascoux polynomials, the $K$-theoretic Demazure characters.

Grothendieck inclusion–exclusion positivity conjecture. For every permutation $w\in S_n$ and every subword $u$ of $w$,

Dennin's conjecture. For every permutation $w$,

Mészáros–Tanjaya's conjecture. For any permutation $w$ and subword $u$ of $w$,

Almost vexillary support conjecture. For every almost vexillary permutation $w$, there exists a subset $A\subset D(w)$ such that

Hafner–Mészáros–Setiabrata–St. Dizier conjecture. For any vexillary permutation $w$, $\hat{\mathfrak{G}}_w$ equals an integer multiple of $\chi_{D_{\mathrm{top}}(w)}$.

Mészáros–St. Dizier conjecture. The support of each Castelnuovo–Mumford polynomial is M-convex.

Let $\beta$ be a formal variable, let $u,v,w\in S_\infty$, and let $\mathfrak{G}_w(\mathbf{x};\mathbf{y})$ denote the double Grothendieck polynomial. Define the coefficients…

Let $\delta_n=(n,n-1,\ldots,1)$ be the staircase partition, and let $\rho\subseteq\delta_n$. Write $\rho'$ for the conjugate partition. Stembridge-type coincidence conjecture. One…

Let $\Lambda=\bigoplus_{\lambda}\mathbb{Q}(\alpha,\beta)s_{\lambda}(\mathbf{x})$ be the ring of symmetric functions over the field of rational functions in $\alpha$ and $\beta$, an…

Bruhat-chain characterization. The coefficient $a_{x,w}$ is nonzero if and only if there exists a saturated chain in the Bruhat order from $w$ to $x$ such that every covering relat…