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YCor
YCor
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Irreducible representations of $SL_n$\mathrm{SL}_n(K)$, $K$ finite

Let $SL_n/K$$\mathrm{SL}_n/K$ ($K$ finite) be given with its natural action on an $n$-dimensional vector space $V/K$. Consider the action of $SL_n$$\mathrm{SL}_n$ on the $m$-fold tensor product $V\otimes \dotsc \otimes V$. What is the largest-dimensional irreducible subrepresentation?

(For $m$ much larger than the maximum of $n$ and the size of $K$, it cannot be the symmetric power, as its dimension becomes larger than the size of $SL_n(K)$$\mathrm{SL}_n(K)$. Or am I missing something?)

Irreducible representations of $SL_n(K)$, $K$ finite

Let $SL_n/K$ ($K$ finite) be given with its natural action on an $n$-dimensional vector space $V/K$. Consider the action of $SL_n$ on the $m$-fold tensor product $V\otimes \dotsc \otimes V$. What is the largest-dimensional irreducible subrepresentation?

(For $m$ much larger than the maximum of $n$ and the size of $K$, it cannot be the symmetric power, as its dimension becomes larger than the size of $SL_n(K)$. Or am I missing something?)

Irreducible representations of $\mathrm{SL}_n(K)$, $K$ finite

Let $\mathrm{SL}_n/K$ ($K$ finite) be given with its natural action on an $n$-dimensional vector space $V/K$. Consider the action of $\mathrm{SL}_n$ on the $m$-fold tensor product $V\otimes \dotsc \otimes V$. What is the largest-dimensional irreducible subrepresentation?

(For $m$ much larger than the maximum of $n$ and the size of $K$, it cannot be the symmetric power, as its dimension becomes larger than the size of $\mathrm{SL}_n(K)$. Or am I missing something?)

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H A Helfgott
H A Helfgott
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Irreducible representations of $SL_n(K)$, $K$ finite

Let $SL_n/K$ ($K$ finite) be given with its natural action on an $n$-dimensional vector space $V/K$. Consider the action of $SL_n$ on the $m$-fold tensor product $V\otimes \dotsc \otimes V$. What is the largest-dimensional irreducible subrepresentation?

(For $m$ much larger than the maximum of $n$ and the size of $K$, it cannot be the symmetric power, as its dimension becomes larger than the size of $SL_n(K)$. Or am I missing something?)