32 problems

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 $k$ be a positive integer, let $\lambda$ be a partition of rank $k$, and let $G_\lambda$ be the associated Giambelli matrix. For any partition $\nu$ of $k$, let…

Let $\lambda/\mu$ be a border strip, let $\theta$ be a partition of the same size, and let $E_{\lambda/\mu}^{\theta}$ be the polynomial associated to the Jacobi–Trudi matrix…

Let $M=(E,\mathcal{I})$ be a matroid with $|E|=n$ and rank $r_M$. For $0\leq k\leq r_M$, define the homogeneous multivariate independent-set polynomial … For polynomials in…

Let $G$ be a graph with vertex set $V$, and for each $v\in V$ let $S(v)$ be its list of available colors. Let $p_\varnothing$ be the polynomial recursively defined from the associa…

Let $d,n$ be positive integers. Write ${\mathbb P}\operatorname{L}(d,n)_{\boxtimes}$ for the space of square-free Lorentzian polynomials of degree $d$ in $n$ variables modulo posit…

Let $H$ be a graph, let $G$ be a graph, and let $h_H(\mathbf{x};G)$ denote the $G$-chromatic function of $H$. Call $G$ antiferromagnetic when it satisfies the antiferromagnetic con…

Let $d\geq 3$, and let $(p_{ij})$ be a real symmetric $d\times d$ matrix with nonnegative off-diagonal entries and zero diagonal entries. For $i\ne j$, let…

Let $\mathfrak{g} = \oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$ be a direct sum of special linear Lie algebras, with positive roots $\Delta$ as above, and let…

Let $f_{A_1,A_2,A_3}$ be the volume polynomial associated with three convex bodies $A_1,A_2,A_3$ in dimension three. A homogeneous polynomial is strongly log-concave when it has th…

Let $M$ be a matroid, and let $\A(M)$ denote the Gorenstein ring associated to its basis-generating polynomial. The Kähler-package conjecture. The ring $\A(M)$ satisfies the Kähler…

Let $M$ be a matroid of rank $r$, and let $\\A(M)$ denote the Gorenstein ring associated to its basis-generating polynomial. The Murai–Nagaoka–Yazawa conjecture. The ring satisfies…

Let $f=\sum_{\alpha\in\mathbb{N}^n}c_{\alpha}x^{\alpha}$ be a polynomial, and define its normalization by … For $\lambda\in \mathrm{Par}^k(d)$, let $\mathfrak{S}_{\lambda}^{(k)}$ d…

Gurvits's conjecture. The equality

Hodge–Riemann and equality conjecture. If all convex bodies in the tuples $E_i$ are smooth and strictly convex, then the valuation

Let $Z$ be a closed subscheme of … and let $I$ be a generating set for its ideal. Let $\zeta_I(t_0,\dots,t_\ell)$ denote the associated Segre zeta function, and consider the numera…

Let $\mathrm{D}$ be a delta-matroid, and let $U_{\mathrm{D}}$ denote its $U$-polynomial. Say that $\mathrm{D}$ has an enveloping matroid when it satisfies the hypothesis used in th…

Let $d$ be a Dyck path, let $G(d)$ be its indifference graph, and let $X_{G(d)}$ be its chromatic symmetric function. For any finite number of variables, restrict $X_{G(d)}$ to tho…

Let $\mathbb{P}\mathrm{L}^d_n$ denote the projective space of all Lorentzian homogeneous polynomials of degree $d$ in the variables $w_1,\ldots,w_n$. The space is compact and contr…

Let $\lambda$ be a partition whose nonzero parts are distinct, and let $P_\lambda$ be its Schur $P$-function. Schur-$P$ conjecture. The normalized function … is Lorentzian for ever…

Let $\lambda,\mu\in\mathbb{Z}_{\geq0}^n$ be partitions satisfying $\lambda_i\geq\mu_i$ for every $i$, and let $s_{\lambda/\mu}$ be the skew Schur polynomial. Skew-Schur conjecture.…

Let $\widetilde{\mathfrak{G}}_w(\mathbf{x},\mathbf{y},z)$ denote the homogenized double Grothendieck polynomial associated with $w\in S_n$. Double-Grothendieck conjecture. The norm…

Let $w\in S_n$ be a permutation and let $\widetilde{\mathfrak{G}}_w(\mathbf{x},z)$ be the homogenized Grothendieck polynomial defined in the source from the homogeneous components…

Let $w\in S_n$ be a permutation, let $\ell(w)$ be its length, and let $\mathfrak{G}_w(\mathbf{1}-\mathbf{x})$ be the Grothendieck polynomial evaluated at $\mathbf{1}-\mathbf{x}$. F…

Let $\alpha\in\mathbb{N}^n$ be a composition, and let $\kappa_\alpha(x_1,\ldots,x_n)$ be its key polynomial. Key-polynomial conjecture. The normalized polynomial … is Lorentzian fo…