62 problems

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

Let $\mathcal{A}$ be the Steenrod algebra and let $\mathcal P_n$ be the polynomial algebra on $n$ degree-one generators over $\mathbb F_2$. For a degree $d$, let $\alpha(d)$ denote…

Richman's conjecture. The elements $x_i$, $N_i$, $u_{ij}$, and $\operatorname{tr}(A)$ for $0\leq a_i\leq p-1$ generate the invariant ring ${\bf F}[mV_2]^{C_p}$. Richman proved the…

Let $\mathbb F$ be an algebraically closed field of positive characteristic, let $\mathbb A$ be the coefficient ring used for reduction modulo the characteristic, and let…

The paper considers Weyl modules for hyper loop algebras over an algebraically closed field of positive characteristic, together with their analogues obtained from characteristic-z…

Let $W$ be the Weyl group, let $w\in W$, and let $Y_x(\dot w)$ be the stratum defined by … For a $G$-regular linear character $\psi:U_0\to\mathcal{O}^{\times}$, let…

Let $W$ be the Weyl group, let $w\in W$, let $\dot w$ be a representative of $w$, and let $\psi:U_0\to\mathcal{O}^{\times}$ be a regular linear character. Torus restriction conject…

Let ${\mathbf{G}}$ be a finite reductive group with Frobenius endomorphism $F$, let ${\mathbf{P}}$ be a parabolic subgroup with an $F$-stable Levi complement ${\mathbf{L}}$, and le…

Top-degree conjecture. For $n>3$,

Let $V$ be a crystalline representation of ${\rm G}_{{\mathbb Q}_p}$, let $T$ be a Galois-invariant lattice, and write $V\bmod p$ for the semisimplification of $T/pT$. Let $\Pi(V)$…

The p-modular TQFT structure conjecture.

Let $\Gamma(1)=SL(2,\mathbb{Z})$ be the modular group, and let the kernel be the normal subgroup of $\Gamma(1)$ whose elements are represented by the identity matrix in the modular…

Let ${\cal Q}_5={\cal W}_5/{\cal U}_5$ and ${\cal U}_5\subset{}\cal W_5$ be the homological quotient and sub-TQFT over $\mathbb Z/5$, and let $Q$ and $U$ be the linear polynomials…

Let $G_n$ and $G_{n-1}$ be the symplectic groups in the preceding branching-rule setting, let $\omega_i^n$ be the relevant fundamental module, and let $D$ be the module occurring i…

Let $p_+$ and $p_-$ be the parameters of the logarithmic $(p_+,p_-)$ model. Let ${\bar{\mathscr{C}}}$ be the space of generalized characters, let $\mathscr{G}$ be the subspace desc…

Let $H$ be a factorizable ribbon quantum group, with ribbon element having multiplicative Jordan decomposition. The decomposition gives $SL(2,\mathbb{Z})$-representations $\bar\pi$…

Folklore irreducibility conjecture. The reduction $\bar{V}_{k,a_p}$ is irreducible for fractional slopes if $k$ is even.

Two-generator conjecture. The ring $\operatorname{Sym}(V^*)^G$ is generated by at most two elements as an algebra over the Steenrod algebra.

Geck's conjecture. The decomposition matrix of the finite group $\mathbf{G}^{\mathrm{F}}$ is upper triangular provided $p$ and $\ell$ are not too small.

Let $k$ be a global field, let $\ell$ be a prime different from the characteristic of $k$, and let $\iota:\mathbb{C}\to\overline{\mathbb{Q}}_\ell$ be a field isomorphism. Let…

Let $p$ be prime, let $V_r$ be the irreducible representation indexed by $r$, and let $\operatorname{Sym}^2(V_r)$ and $\operatorname{Alt}^2(V_r)$ denote its symmetric and alternati…

Let $M,N\in\mathbb{N}$, let $d\in\mathbb{N}_0$, let $\mathbb{F}$ be the underlying field, and let $E$ be the natural $2$-dimensional $\mathbb{F}$-representation of…

Square-irreducibility conjecture. $\mathrm{Z}(\mathfrak{m})$ is square-irreducible if and only if $\mathcal{C}_\mathfrak{m}^+$ contains an open $\mathrm{GL}(V)$-orbit.

Meyer and Neutsch's conjecture. The third candidate should be the unique irreducible Brauer character of $Ly$ of degree $111$.

Let $\ell$ be a prime and let $q$ be a prime power. The naive converse theorem for mod $\ell$ representations of $\operatorname{GL}_2(\mathbb{F}_q)$ asks whether the relevant mod…