2,491 problems

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…

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…

For admissible parameters $a,b$ and any two lattice paths $P,Q\in\mathcal{D}(a,b)$, does equality of their Lagrange numbers imply that their associated band graphs are isomorphic?…

For every connected complex reductive algebraic group $G$ and every involutive automorphism $\theta:G\to G$, let $H=G^\theta$ and let $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{s}$…

For every finite-dimensional representation-finite algebra $A$ and every integer $i\ge 0$, the number of quotient-closed full additive subcategories…

For every finite-dimensional associative algebra $A$ over a field $K$, the quotient of its Hochschild cohomology ring by the ideal of nilpotent elements is a finitely generated $K$…

For a $3$-dimensional $\mathcal{N}=4$ gauge theory with Higgs branch $\mathcal{M}_H$, dual Coulomb branch $\mathcal{M}_C$, and quantized Coulomb-branch algebra $\mathcal{A}_C$, the…

For every finite simple graph $G$ with distinguished vertex $v$, let $\operatorname{Aut}_v(G)\leq \operatorname{Aut}(G)$ be the subgroup fixing $v$. Then the conjecture asserts tha…

For every finite group $G$ and prime $p$, let $P\in\operatorname{Syl}_p(G)$ be nonabelian, and define $m(P)=\min\{\varphi(1):\varphi\in\operatorname{Irr}(P),\ \varphi(1)>1\}$. Then…

For every finite group $G$, every prime $p$, and every Sylow $p$-subgroup $P\in\operatorname{Syl}_p(G)$, define…

For every complex simple Lie algebra $\mathfrak{g}$, with $\rho$ the half-sum of its positive roots, and every dominant integral weight $\lambda\in P_+$, the irreducible representa…

For Hermitian matrices $A_{1},A_{2}\in M_{n}(\mathbb{C})$, define $p(z_{0},z_{1},z_{2})=\det\!\left(z_{0}I+z_{1}A_{1}+z_{2}A_{2}\right)$. Kippenhahn's conjecture asserts that if th…

The PBW filtration conjecture. One has

Let ${\mathfrak{g}}$ be a simple basic classical Lie superalgebra, and let $L(\lambda)$ be a finite-dimensional simple ${\mathfrak{g}}$-supermodule. Write…

Terwilliger systems' subdegree bound conjecture. The inequality above holds for every TD system.

Generalized Amdeberhan–Shareshian–Stanley conjecture. For every partition $\mu$ of $kn$, the symmetric function obtained by applying the partition division map…

Let $G_r$ be an unramified base-change form of the quasi-split group $G$, with Hecke algebras $\mathcal{H}(G_r)$ and $\mathcal{H}(G)$, and let $\mathfrak{Z}^{\textnormal{st}}(G_r)$…

Let $C^{(l+1,1)}(\alpha)$, $C^{(l+1,1)^T}(\alpha)$, $P_{l+4+k}^{(l+1,1)}$, and $P_{l+4+k}^{(l+1,1)^T}$ denote the polynomials and Chebyshev series defined in the preceding formulae…

Picky Conjecture. There exists a bijection

Let $\mathcal{A}_{\Gamma_\infty'}$ be the cluster algebra associated with the quiver $\Gamma_\infty'$, and let $\mathcal{O}^{sh}$ be the shifted category appearing in the isomorphi…

Let $(pi,pi')$ be a coherent pair with Harish-Chandra parameters $\lambda$ and $\lambda'$ sufficiently far from the walls. Let $(\sigma,\sigma')$ be an elementary pair, let $F$ and…

Let $R_n^{(1,2)}$ be the $(1,2)$-bosonic-fermionic coinvariant ring, and let $\operatorname{Frob}(R_n^{(1,2)};q;u,v)$ denote its multigraded Frobenius series. The notation…

Let $R_n^{(2,2)}$ be the $(2,2)$-bosonic-fermionic coinvariant ring, and let $\operatorname{Frob}(R_n^{(2,2)};q,t;u,v)$ denote its multigraded Frobenius series. Here $\Theta_{e_k}$…

Let $G$ be the ambient reductive group, let $P$ be a parabolic subgroup of $G$ containing $B$, with Lie algebra $\mathfrak{p}$ and opposite parabolic $\overline{P}$. Let $M$ be a…