Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $Y =
\operatorname{Sp}(V)$.

35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph