Unipotent elements forcing irreducibility in linear algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if $u$ is a regular unipotent element, then $X$ cannot be contained in a proper parabolic subgroup of $G$. We generalize their result and show that if $u$ has order $p$, then except for two known examples which occur in the case $(G, p) = (C_2, 2)$, the subgroup $X$ cannot be contained in a proper parabolic subgroup of $G$. In the case where $u$ has order $> p$, we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.Comment: 33 page

Invariant forms on irreducible modules of simple algebraic groups

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$ and let $V$ be an irreducible rational $G$-module with highest weight $\lambda$. When $V$ is self-dual, a basic question to ask is whether $V$ has a non-degenerate $G$-invariant alternating bilinear form or a non-degenerate $G$-invariant quadratic form. If $p \neq 2$, the answer is well known and easily described in terms of $\lambda$. In the case where $p = 2$, we know that if $V$ is self-dual, it always has a non-degenerate $G$-invariant alternating bilinear form. However, determining when $V$ has a non-degenerate $G$-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where $G$ is of classical type and $\lambda$ is a fundamental highest weight $\omega_i$, and in the case where $G$ is of type $A_l$ and $\lambda = \omega_r + \omega_s$ for $1 \leq r < s \leq l$. We also give a solution in some specific cases when $G$ is of exceptional type. As an application of our results, we refine Seitz's $1987$ description of maximal subgroups of simple algebraic groups of classical type. One consequence of this is the following result. If $X < Y < \operatorname{SL}(V)$ are simple algebraic groups and $V \downarrow X$ is irreducible, then one of the following holds: (1) $V \downarrow Y$ is not self-dual; (2) both or neither of the modules $V \downarrow Y$ and $V \downarrow X$ have a non-degenerate invariant quadratic form; (3) $p = 2$, $X = \operatorname{SO}(V)$, and $Y = \operatorname{Sp}(V)$.Comment: 46 pages; to appear in J. Algebr

Orthogonal irreducible representations of finite solvable groups in odd dimension

We prove that if $G$ is a finite irreducible solvable subgroup of an orthogonal group $O(V,Q)$ with $\dim V$ odd, then $G$ preserves an orthogonal decomposition of $V$ into $1$-spaces. In particular $G$ is monomial. This generalizes a theorem of Rod Gow.Comment: to appear in Bull. Lond. Math. So

Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known.Comment: to appear in Proc. Amer. Math. So

Adjoint Jordan blocks for simple algebraic groups of type CℓC_{\ell} in characteristic two

Let $G$ be a simple algebraic group over an algebraically closed field $K$ with Lie algebra $\mathfrak{g}$. For unipotent elements $u \in G$ and nilpotent elements $e \in \mathfrak{g}$, the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ are known in most cases. In the cases that remain, the group $G$ is of classical type in bad characteristic, so $\operatorname{char} K = 2$ and $G$ is of type $B_{\ell}$, $C_{\ell}$, or $D_{\ell}$. In this paper, we consider the case where $G$ is of type $C_{\ell}$ and $\operatorname{char} K = 2$. As our main result, we determine the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ for all unipotent $u \in G$ and nilpotent $e \in \mathfrak{g}$. In the case where $G$ is of adjoint type, we will also describe the Jordan block sizes on $[\mathfrak{g}, \mathfrak{g}]$.Comment: 39 page

Fast Monotone Summation over Disjoint Sets

We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size $p$ of an $n$-element ground set. More precisely, the task is to compute, for each subset of size $q$ of the ground set, the sum over the values of all subsets of size $p$ that are disjoint from the subset of size $q$. We present an arithmetic circuit that, without subtraction, solves the problem using $O((n^p+n^q)\log n)$ arithmetic gates, all monotone; for constant $p$, $q$ this is within the factor $\log n$ of the optimal. The circuit design is based on viewing the summation as a "set nucleation" task and using a tree-projection approach to implement the nucleation. Applications include improved algorithms for counting heaviest $k$-paths in a weighted graph, computing permanents of rectangular matrices, and dynamic feature selection in machine learning