118 problems

Functional equation conjecture. The exponential generating function satisfies

For every positive integer $k$ and every $k$-bounded partition $\lambda$, the $K$-$k$-Schur function $\mathsf{KSchur}^{(k)}_{\lambda}$ is a nonnegative integer linear combination o…

For every finite connected weighted graph $G=(V,E,w)$, the spectral gap of the interchange process on $G$ equals the spectral gap of the continuous-time random walk on $G$:…

Kirillov's conjecture. For every $u,v,w\in S_n$ and every integer $N\geq 1$,

Consider the Grassmannian $\limfunc{Gr}_{n,N}$ and the ring … For partitions ${\Greekmath 0115}$ and ${\Greekmath 0116}$ indexing opposite Schubert varieties and fixed points, let…

Alternating-sign conjecture. For every $u,v,w\in W$,

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 $m\in\mathbb Z_{>0}$, let $\mathcal C\subseteq\mathsf{RPD}_{m,n}$ be a $\sqrt{\mathfrak{gl}_n}$-crystal, and let $w\in S_n$. Write … A polynomial is Lascoux positive if it is a…

Kac–Moody extension conjecture. The analogues of the equations giving the Chevalley formulae for $\lambda$ and $-\lambda$ hold for $W$.

Equivariant positivity conjecture. The coefficient $c_{v;w}(\alpha)$ is a polynomial in the positive roots $\alpha$ with non-negative coefficients. Numerical evidence supports this…

Let $m,p\geq 1$, let $\gamma$ be the real rational normal curve, and let $w_1,\ldots,w_n$ be Schubert conditions on $p$-planes in $(m+p)$-space. For points…

Let $\alpha$, $\beta$, and $\gamma$ be strings indexing Schubert classes in the two-step flag variety, and let $c^{\gamma}_{\alpha,\beta}$ denote the corresponding two-step Littlew…

Nonnegativity conjecture. For all $u,v,w$, $c_{uv}^w(y;z)$ is a polynomial in the differences $y_i-z_j$ with nonnegative integer coefficients.

Let $\omega$ be the permutation governing the switching algorithm, and let $\mathcal{O}\in\operatorname{Orb}(\omega)$ be one of its orbits. A genomic tableau is a tableau counted b…

Cosmall-root characterization conjecture. Assume that $R$ is simply laced and let $\alpha\in R^+\setminus R^{+}_P$. Then $\alpha$ is $P$-cosmall if and only if

Positivity conjecture for motivic Chern classes. For every $u\leq w\in W$,

Alternation conjecture. The Euler characteristic is alternating:

Strong positivity conjecture. One has $c_{v,w}>0$ for every $v\le w$. The weaker non-negativity assertion is known, but strict positivity remains the stronger conjectural statement…

Let $R_n$ be the ring defined in the surrounding construction, with binary operation $\star$. Associativity conjecture. $(R_n,\star)$ is an associative ring. The supplied context d…

Let $\widetilde{\cal G}_n^0$ be the quotient of $\widetilde{\cal G}_n$ by the relations $[ij]^2=0$. Let $\widetilde\theta_1,\ldots,\widetilde\theta_n$ be the Dunkl elements, and le…

Let $S_n$ be the symmetric group, let $\mathcal{E}_n^+$ be the positive cone generated by noncommutative monomials in the Bruhat generators $[ij]$, and let…

Let $m,p\geq 1$, let $\gamma$ be the real rational normal curve, and let $w_1,\ldots,w_n$ be Schubert conditions satisfying … For distinct real points $s_1,\ldots,s_n\in\gamma$, co…

Vakil's geometric Littlewood–Richardson conjecture. There exists a subset $U\subseteq S\times S\times\{d_0,\dots,d_{\binom n2}\}$ such that: $(\alpha,\beta,d_0)\in U$ for all…

Let $X_1$ and $X_2$ be the two components arising when a puzzle variety breaks in the geometric Littlewood–Richardson degeneration, and let $X_3$ be the puzzle variety correspondin…

Let $0\le d<n$, let $t_1<t_2<\dots<t_{(n+1)(d-n)}$ be real numbers, and let $\mathbf r:\mathbb C\to\mathbb C^{d+1}$ be a rational normal curve with coordinates $r_i(t)=t^i$ for…