25 problems

The branching coefficient formula. The coefficients are

Height-three induction conjecture.

Restriction branching conjecture. Except for $A$, the only $\lambda^A$-normal node of residue $\alpha$ is $B$, and

Induction branching conjecture. There are exactly two such conormal nodes, and

Restriction branching conjecture. The result for the restriction functor $\rho_\beta$ dual to Lemma 2.5 should also hold, and should follow from the reciprocal result for the induc…

Let $\mathfrak{h}\subset\mathfrak{sl}(m|n)$ be a subalgebra specified by a subset of positive roots $\Psi^+$, with decomposition $\Psi^+=\Psi_0^+\sqcup\Psi_1^+$ into even and odd r…

Let $P$ and $\leftindex^L{P}$ be the weight lattices of $\mathfrak{so}_{2n+1}$ and $\mathfrak{sp}_{2n}$, respectively, and let $\Pi$ be the character map described in the paper. Fo…

Let $n$ be a positive integer. Consider the restriction of finite-dimensional, irreducible polynomial representations of $GL_{2n}(\mathbb{C})$ to $Sp_{2n}(\mathbb{C})$, together wi…

Parabolic branching polynomial positivity conjecture. The polynomial $b_{\nu,\overline{\lambda}+\gamma}(p)$ has nonnegative integer coefficients at least when…

Let $G$ be a simply connected complex algebraic group with Lie algebra $mathop{mathrm{Lie}}G=mathfrak{g}$, let $U$ be a maximal unipotent subgroup of $G$, and set … For a subalgeb…

Let $k\in\mathbb{N}$ and let $\lambda\vdash 2^k$. Write $l(\lambda)$ for the number of parts of $\lambda$, let $\lambda_{l(\lambda)}$ be its smallest part, and let $Z^\lambda$ be t…

Let $s=(s_1,\ldots,s_n)$, let $\theta=\{\theta_{ij}\}_{1\leqslant i<j\leqslant n}$ be nonnegative integers, and define … The branching coefficients $e_n(s;\theta;q,t)$ are defined…

The $(\mathrm{C}_n,\mathrm{B}_n)$ branching-rule conjecture. The two displayed identities hold.

Finite generation conjecture. The branching semigroup $\Gamma$ is always finitely generated in this context, as well as in the less restrictive context considered by the cited auth…

Let $\sigma$ be an involution of ${\mathfrak g}$, let $H_1,\ldots,H_l$ be the chosen basic invariants with degrees $d_1,\ldots,d_l$, and let $\varpi$ be the symmetrisation map. Let…

Let $\mathbb{B}$ range over Littelmann path models for the irreducible representation $\operatorname{L}(\lambda)$, and let the branching property mean that the theorem asserting th…

Let $\mathfrak{g}=\mathfrak{sl}(2n,\mathbb{C})$, let $\mathfrak{g}^{\sigma}=\mathfrak{sp}(2n,\mathbb{C})$ be the fixed-point subalgebra, and let $\mathbb{B}$ be the path model for…

Generalized Macmahon branching conjecture. The module $\mathcal M_{\alpha,\beta,\emptyset}^{(n;0)}(u,K)$ should admit the displayed decomposition as an…

Let $\mathfrak{g}$ be a simple Lie algebra, let $\mathfrak{a}\subset\mathfrak{g}$ be a simple Levi subalgebra, and let $\lambda$ be a dominant integral weight for $\mathfrak{g}$ su…

Let ${\mathfrak g}$ be a semisimple Lie algebra, let $i(\bar{{\mathfrak g}})$ be reductive in ${\mathfrak g}$, and assume that…

Let ${\mathfrak g}$ be a semisimple Lie algebra, let $i(\bar{{\mathfrak g}})$ be reductive in ${\mathfrak g}$, and suppose that…

Associated-variety equality conjecture. The inclusion in that theorem is equality:

Discrete decomposability equivalence conjecture. The following two conditions on $(G,G',\pi)$ are equivalent: (i) the restriction $\pi|_{G'}$ is infinitesimally discretely decompos…

Virtually symmetric-type bounded multiplicity conjecture. If $\mathfrak{q}$ is of virtually symmetric type, then the restriction

Let $q$ be the cardinality of the residue field, and let $V_{(2,3,4)}$, $V_{(1,3,4)}$, $V_{(2,2,4)}$, and $V_{(3,2,3)}$ be the representations indexed by the indicated triples in t…