2,594 problems

Let $n\geq 2$ and let $\Gamma$ be a subgroup of $\mathbf{SL}_n\mathbb{C}$. Let $\{\gamma_0,\ldots,\gamma_l\}$ be the equivalence classes of irreducible finite-dimensional complex r…

Let $P(U_1,\ldots,U_k,U_1^*,\ldots,U_k^*)$ be a noncommutative nonzero $*$-polynomial with complex coefficients, let $\underline{\lambda}\neq 0$ be a signature, and let…

The universal-enveloping-algebra expression conjecture. For every $n$,

For $\alpha\in Q^{\geq 0}$, let $\mathfrak{J}_\alpha$ be the Toda eigenfunction and let $(q)_\alpha$ be the associated $q$-factorial. A polynomial is unimodal if its coefficient se…

Prove the six affine Jacobi–Trudi identities conjectured by Ole Warnaar for the relevant specialisations of multiparameter Hall–Littlewood polynomials of types $\mathrm{B}_n$,…

Let $X$ be a rationally smooth complex affine variety with an algebraic torus action, and let $x\in X$ be an attractive fixed point. Let $p$ be a prime, assume that…

Let $\mathfrak{g}$ be the rank-three hyperbolic Kac–Moody Lie algebra of type $A_1^{++}$, with invariant bilinear form $(\cdot,\cdot)$, and let $p(n)$ denote the ordinary partition…

For each $n\geq 4$, let $T_n=\{\mathrm{id}\}\cup\{(i\ j):1\leq i<j\leq n\}\subseteq S_n$. Is it true that there is no subset $A\subseteq S_n$ such that the multiplication map…

Let $W_{\geq -1}=\mathbb{C}[t]\partial$ and $W=\mathbb{C}[t,t^{-1}]\partial$ be the Witt Lie algebras. The conjecture is that the enveloping algebras…

For every integer $n\ge 1$, every integer $1\le r\le n$, every real matrix $A=(a_{ij})\in\mathbb{R}^{n\times n}$ whose minors of orders at most $r$ are nonnegative, and every parti…

Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$, and let…

For every $n\ge 1$ and every pair of weak compositions $\alpha,\beta\in\mathbb{Z}_{\ge 0}^n$, the product of key polynomials admits a nonnegative expansion in the basis of Demazure…

For every finite-dimensional quantum complete intersection $A$ over an arbitrary field and every finite-dimensional left $A$-module $M$, if $\operatorname{Ext}^{1}_{A}(M,M)=0$ and…

For every finite group $G$ and every field $K$, if $\operatorname{Ext}^1_{KG}(K,K)\neq 0$, where $K$ is regarded as the trivial $KG$-module, then…

Let $G$ be a simple Lie group of real rank $r\ge 2$, let $\Gamma<G$ be a lattice, and let $X$ be an acyclic simplicial complex with $\dim X<r$. Then every simplicial action of…

Let $G_n$ be an even orthogonal group over a non-Archimedean local field and let $\psi$ be a local Arthur parameter for $G_n$. Determine the local Arthur packet $\Pi_\psi(G_n)$ and…

Let $A$ be a finite-dimensional algebra over a field. Assume that $A$ satisfies the Auslander condition: for every finitely generated left $A$-module $M$, there exists an integer…

Let $G$ and $A$ be finite groups such that $A$ acts on $G$ by automorphisms and $\gcd(|A|,|G|)=1$. For every prime $p$, the number of $A$-invariant irreducible $p$-Brauer character…

Let $H(C,D,\Omega)$ be a Geiß–Leclerc–Schröer algebra associated with a symmetrizable Cartan matrix $C$, symmetrizer $D$, and orientation $\Omega$. Let $\Delta^+(C)$ denote the set…

Let $\mathpzc K$ be a field of characteristic $0$ and let $\varepsilon\in\mathpzc K^{\times}$ be not a root of unity. For every integer $n\geq 2$ and every integer $r\geq 0$, the n…

For each integer $n\geq 3$, let $G=\mathrm{SL}_n(\mathbb{Z})$ and, for a finite generating set $S\subseteq G$, let $\kappa(G,S)$ denote its Kazhdan constant. Is…

Let $A$ be a finite-dimensional algebra over a field $K$, and let $\mathcal{G}\subseteq \mathrm{HH}^*(A)$ be the Gerstenhaber ideal generated by all homogeneous nilpotent elements…

For every pair of coprime integers $1<a<b$, the matrix-counting $q$-series $Z_{a,b}(q)$ equals the explicit infinite product $P_{a,b}(q)$, namely $Z_{a,b}(q)=P_{a,b}(q)$. Equivalen…

Let $G$ be a Kunze--Stein locally compact group, let $\lambda_G$ denote its left regular representation, and let $\pi$ be a unitary representation of $G$ such that…

Let $H_n(q)$ be the generic Iwahori--Hecke algebra of the symmetric group $d504_n$ with standard basis $T_w$ indexed by $w\in \mathfrak{S}_n$. For $1\leq i\leq \frac{n(n-1)}{2}$, d…