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Rational modules need not be semisimple in characteristic p

Statement refuted

Assume the Axiom of Choice inherited from the named suppliers. The claim that every finite-dimensional rational representation of a split reductive group is semisimple is false. For every prime $p$ the following is a counterexample (Complete reducibility of rational modules in characteristic zero). Let $k$ be a field of characteristic $p$ and let $G = {S L}_{2}$ act on $V = S^{p} (k^{2})$ , the symmetric power of the standard two-dimensional representation, with $T_{2}$ acting through the characters $p, p - 2, \dots, - p$ (Structure of SL_2 and root coordinates, The simple modules of SL_2 and its fundamental representation). Let $W \subseteq V$ be the span of $e_{1}^{p}$ and $e_{2}^{p}$ . Then $W$ is a two-dimensional simple submodule (the Frobenius twist of $L (1)$ ), every simple submodule of $V$ equals $W$ , so the socle of $V$ is $W$ , and $\dim_{k} V = p + 1 > 2$ . Hence $V$ is a nonsemisimple finite-dimensional rational representation of the split reductive group ${S L}_{2}$ , even though the dominant weights still classify the simple rational representations (Dominant weights classify the simple rational representations of a split reductive group).

Facts & Assumptions

Given: AC; a prime $p$ , a field $k$ of characteristic $p$ , $G = {S L}_{2}$ with diagonal torus $T_{2}$ and upper unipotent group $U^{+}$ , and $V = S^{p} (k^{2})$ with its standard basis monomials $e_{1}^{a} e_{2}^{p - a}$ .

[F1]

The Frobenius-twist submodule. $W = k e_{1}^{p} \oplus k e_{2}^{p}$ is a two-dimensional simple $G$ -submodule of $V$ isomorphic to $L (p)$ , with $W_{p χ} = k e_{1}^{p}$ ; it is the Frobenius twist of $L (1) = k^{2}$ , and $e_{1}^{p}$ generates $W$ as a $G$ -module (The simple modules of SL_2 and its fundamental representation).

[F2]

Primitive vectors of simple modules. Every simple rational representation of $G$ contains a primitive vector, unique up to a nonzero scalar, whose weight is its highest weight (Simple rational representations have a unique highest weight, Primitive vectors for a Borel pair).

[F3]

Weights and the unipotent action on $V$ . The monomials $e_{1}^{a} e_{2}^{p - a}$ ( $0 \leq a \leq p$ ) form a $k$ -basis of $V$ of $T_{2}$ -eigenvectors, with $e_{1}^{a} e_{2}^{p - a}$ of weight $(2 a - p) χ = (p - 2 (p - a)) χ$ , and $u_{α} (t) \cdot e_{1}^{a} e_{2}^{p - a} = e_{1}^{a} (e_{2} + t e_{1})^{p - a}$ for $t \in k$ , where $u_{α} (t) = (\binom{1 t}{0 1})$ acts by $e_{1} \mapsto e_{1}$ and $e_{2} \mapsto e_{2} + t e_{1}$ (Structure of SL_2 and root coordinates, Symmetric algebra of a vector space, Weights, dominant weights and the highest-weight order of a rational representation, Rational representations and comodules of an affine group scheme).

[F4]

Classification. The simple rational representations of $G$ are classified up to isomorphism by their dominant highest weights (Dominant weights classify the simple rational representations of a split reductive group).

Counterexample

Proof technique: direct.

1.1F3

The basis monomials $e_{1}^{a} e_{2}^{p - a}$ of $V$ are $T_{2}$ -eigenvectors of weights $(2 a - p) χ$ for $a = 0, \dots, p$ , i.e. of the characters $- p, - p + 2, \dots, p$ ; each weight space of $V$ is therefore one-dimensional and spanned by a single monomial, and $T_{2}$ acts through the characters $p, p - 2, \dots, - p$ as claimed.

2.1F3givenalgebra

The primitive vectors of $V$ are exactly the nonzero multiples of $e_{1}^{p}$ . Indeed, a primitive vector is a nonzero $T_{2}$ -eigenvector fixed by $U^{+}$ (Unipotent algebraic groups and unipotent representations, Primitive vectors for a Borel pair), hence by step 1.1 is a nonzero multiple of some monomial $m_{a} = e_{1}^{a} e_{2}^{p - a}$ ; and $u_{α} (t) \cdot m_{a} = e_{1}^{a} (e_{2} + t e_{1})^{p - a}$ has, as the coefficient of $e_{1}^{p}$ , the term $t^{p - a}$ (the $i = p - a$ summand, with binomial coefficient $1$ ). If $a < p$ , the original monomial has zero coefficient of $e_{1}^{p}$ , whereas the translated coefficient is the nonzero polynomial $t^{p - a}$ in $k [t]$ . Thus fixedness over every base algebra excludes $a < p$ . For $a = p$ , $u_{α} (t) \cdot e_{1}^{p} = e_{1}^{p}$ for all $t$ , so $e_{1}^{p}$ is fixed.

3.1F1F2step 1.1step 2.1

Let $S \subseteq V$ be a nonzero submodule; if $S$ is simple, then by [F2] it contains a primitive vector $v$ , which by step 2.1 is a nonzero multiple of $e_{1}^{p}$ ; hence $e_{1}^{p} \in S$ . By [F1] the $G$ -submodule generated by $e_{1}^{p}$ is $W$ (it is nonzero and contained in the simple module $W$ ), so $W \subseteq S$ , and simplicity of $S$ gives $S = W$ . Therefore $W$ is the unique simple submodule of $V$ and the socle of $V$ is $W$ . Since $\dim_{k} V = p + 1$ while $\dim_{k} W = 2$ , a semisimple $V$ would be the sum of its simple submodules, namely $W$ , which is impossible; hence $V$ is not semisimple.

4.1F4step 3.1∎

The representation $V = S^{p} (k^{2})$ is finite-dimensional and rational, $G = {S L}_{2}$ is a split reductive group of characteristic $p$ , and $V$ is not semisimple by step 3.1, while its simple submodules and those of every rational $G$ -module are still classified by the dominant weights by [F4]. This refutes the claim that every finite-dimensional rational representation of a split reductive group is semisimple.

Remarks

The failure has the same source as the Frobenius twist of the example: the submodule $W$ generated by $e_{1}^{p}$ is only two-dimensional inside the $(p + 1)$ -dimensional symmetric power, so the top-weight vector does not generate $V$ in characteristic $p$ .
The statement refers to the characteristic-zero theorem Complete reducibility of rational modules in characteristic zero; the counterexample is compatible with the classification theorem, which only parametrizes simple modules and says nothing about extensions.

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Robert Steinberg, Lectures on Chevalley Groups (Yale University, 1967; notes prepared by J. Faulkner and R. Wilson) (standard reference, not scraped)