37 problems

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

Let $G_\mathbb{F}$ be the reductive group over the field $\mathbb{F}$ considered in the paper, and let $T$ be an indecomposable tilting $G_\mathbb{F}$-module. The support variety o…

Let $G$ be a semisimple algebraic group over $k$, let $r\geq 1$, let $M$ be a rational $G$-module, and let $\operatorname{St}_r$ denote the $r$-th Steinberg representation. A good…

Riche–Williamson's p-canonical basis conjecture. The values at $v=1$ of the coefficients ${}^p m_{y,x}$ express the characters of tilting modules in terms of Weyl characters. This…

Let ${\mathcal O}$ be a nilpotent orbit and let $d_{\mathcal O}$ be its distinguished involution. Let $T(d_{\mathcal O}\cdot 0)$ be the corresponding tilting module and let…

Let $G$ be the simply connected group of type $E_6$, and let $H$ be the connected subgroup of fixed points of the Dynkin-diagram involution, of type $F_4$. A pair $(G,H)$ is a Donk…

Let $\mathfrak{gl}(m|n)$ and $\mathfrak{q}(n)$ be the indicated Lie superalgebras, and let an indecomposable tilting module have a $\Delta$-flag, meaning a filtration whose subquot…

Character map conjecture. If $i:K(\mathcal O_{m|n}^\Delta)\to\mathscr T^{m|n}_{\mathbb Z}$ is the map defined in the cited theorem, then

Let $\eta\in Y^+$ and suppose that $\eta-2\check\rho\in Y^+$. Let $K^\bullet_\eta$ be the graded multiplicity space of $\IC(\ol\fM_\eta)$ in $\bigoplus_{\xi\in Y}\bp_*\IC(\fQ^\xi)$…

Tensor-ideal equality conjecture. For any $G$ and $p$, all of the following hold:

Naturality conjecture. The conditions in Proposition are satisfied for an arbitrary $\mathfrak{g}$, with the assumptions of Section; equivalently, support of tilting modules is nat…

Achar–Hardesty–Riche classification conjecture. There is a bijection

Scheme-theoretic traditional Humphreys conjecture. For every $w\in W_{\mathrm{ext}}^\varnothing$, the coherent sheaf

Relative Humphreys conjecture.

Traditional Humphreys conjecture.

Williamson–Riche tilting character conjecture. Replacing the Kazhdan–Lusztig polynomials in Andersen's conjecture by their $p$-versions should give a formula for the characters of…

Let $\Bbbk$ be an algebraically closed field of characteristic $p>0$, let $E\simeq C_p^r$ be an elementary abelian $p$-group, and fix an embedding $E<\Bbbk^+<SL_2(\Bbbk)$. Let…

Let $G$ be the reductive group, $\rho$ the half-sum of positive roots, $\alpha_0^{\vee}$ the coroot associated with the highest short root, and let $\mathbb{X}^+$ denote the domina…

Mazorchuk–Parker's conjecture. The finitistic dimension of $A$ is equal to two times the projective dimension of its characteristic tilting module:

For a nilpotent orbit $C \subset \mathcal{N}$, let $\mathcal{I}_{\overline{C}} \subset \Bbbk[\mathcal{N}]$ denote the defining ideal of the reduced subscheme…

When $\mathfrak{g}=\mathfrak{sp}_4$ and $\ell=3$, let $\mathcal{O}_{\mathrm{subreg}}$ be the subregular nilpotent orbit, choose $u\in\mathcal{O}_{\mathrm{subreg}}$, and let $G_u$ b…

Let $\mathbb{X}^+$ be the dominant weights and let $lambda,mu,mu'\inmathbb{X}^+$. Write … where $s(\mu)$ is the Weyl-orbit sum. Use the order relation $\uparrow$ on weights. Monoto…

Let $G$ be the reductive group, $\mathbb{X}$ its weight lattice, $\mathbb{X}^+$ the dominant weights, and $\mathbb{X}_1$ the restricted weights. For $lambda\rinmathbb{X}_1$ and…

Suppose we are in the reductive case. Let $boldsymbol{ u}in-boldsymbol{X}^+$, and let $({mathscr{O}}{boldsymbol{ u}},boldsymbol{rho}{boldsymbol{ u}})inXi$ be the pair corresponding…

Let $l>2$, let ${\boldsymbol{\lambda}}$ and ${\boldsymbol{\mu}}$ be weights for the level $l$ generalised blob algebra over a field of characteristic $0$, and for each weight let…