5,300 problems

In mathematics, specifically in the field of group theory, the McKay conjecture is a theorem of equality between two numbers: the number of irreducible complex characters of degree…

In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial $P_{y,w}(q)$ is a member of a family of integral polynomials introduced by David Kazhdan and George…

In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by Demazure. The conjecture implies that many of the result…

In finite group theory, Dade's conjecture is a conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups,…

In mathematics, the Arthur conjectures refer to a set of conjectures proposed by James Arthur in 1989. These conjectures pertain to the properties of automorphic representations of…

Lusztig's character formula conjecture. If $p \geq h$ and $w \in W_{\mathrm{aff}}^\varnothing$ satisfies

Let $\psi\in\Psi({}^{\vee}G^{\Gamma})$ be an Arthur parameter, let $\varphi_\psi$ be its associated Langlands parameter, and let $\Pi(G)_\varphi$ denote the corresponding $L$-packe…

Kazhdan–Lusztig positivity conjecture. The coefficients of the Kazhdan–Lusztig polynomials are non-negative, namely $h_{y,x}\in v\mathbb{Z}_{\geq0}[v]$, and the structure constants…

Let $\pi$ be an unramified cuspidal representation of $\operatorname{GL}_N$ and let $\alpha_1,\ldots,\alpha_N$ be its Satake parameters at a prime $\mathfrak{p}$. Generalized Raman…

Irreducibility conjecture. The family of representations $\mathcal{H}_{\bar{\rho}}^+$ is irreducible.

Donovan's conjecture. For a fixed defect group, there are only finitely many block algebras of group-block pairs up to Morita equivalence.

Let $(W,S)$ be the Coxeter system underlying the Soergel bimodule category, let $B_x$ denote the indecomposable Soergel bimodule indexed by $x\in W$, and let $b_x$ be the correspon…

Shahidi's conjecture. Every tempered $L$-packet contains a generic member.

Nakayama's conjecture. If an Artin algebra has infinite dominant dimension, then it is self-injective.

Let $G$ be a reductive group over a nonarchimedean local field $F$, let $\varphi$ be a discrete Langlands parameter, and let $\pi \in \Pi_{\varphi}^2(G,F)$ correspond to an irreduc…

Let $N$ be a positive integer, and let $m,n$ be integers with $m\leqslant n$. For representations of $\operatorname{SL}_N(\mathbb{C})$, write $U\leqslant V$ when $U$ is isomorphic…

Fusion products are constructed from tensor products of highest-weight cyclic modules for a current Lie algebra, using evaluation maps associated with a finite subset of…

Let $C$ be a finite complex reflection group, let $V$ be its reflection representation, and let $H_C(\vec{u})$ be the generic Hecke algebra defined from the braid group of $C$ and…

Let $U_{q}^{7}(\mathfrak{g})$ be an affine quantum group without the degree operator, and let $W^{r,s}$ be a Kirillov–Reshetikhin module, where $r$ is a node of the Dynkin diagram…

Donkin-type tilting conjecture. For every $\lambda\in X_{n-1}(T)$, the tensor product

Braid-group fixed-point conjecture. For every $i\in[d-1]$, if $(k,n)$ is $(3,9)$ or $(4,8)$, every unstable fixed point of $\sigma_i$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$ is a c…

Let $G$ be a real reductive Lie group and let $X$ be a $(\mathfrak{g},K)$-module with Dirac cohomology $H_D(X)$. The infinitesimal character of $X$ is the character by which the ce…

Let $k$ be a field and let $\Lambda$ be a finite-dimensional $k$-algebra with Jacobson radical $\mathfrak{r}$ such that $E=\Lambda/\mathfrak{r}$ is separable. Write…

Let $e$ be the quantum characteristic of a non-trivial invertible element $q\in\mathbb{F}^{\times}$, and let $d_{\lambda,\mu}(v)$ be the integer-coefficient polynomials associated…

Let $R$ and $S$ be artin algebras, and let ${}_RC_S$ be the relevant Wakamatsu tilting bimodule. Write $\mathop{\rm pd}\nolimits_RC$ and $\mathop{\rm pd}\nolimits_{S^{op}}C$ for th…