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irreducible representations of the special linear group over 
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(Theorem)
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Let $p\geq 5$ be a prime and let $G=\SL(2,\mathbb{F}_p)$ be the special linear group over the field with $p$ elements.
Next, we construct several irreducible representations for $G=\SL(2,\F)$ . For $0\leq m \leq p-1$ , let $V_m$ be the $\F$ vector space of homogeneous polynomials of degree $m$ in the independent variables $x$ and $y$ (of course, for $m=0$ , the representation is trivial). We give
$V_m$ a structure of $\F[G]$ -module as follows. Let $p=p((x,y))\in V_m$ and $A\in \SL(2,\F)$ . We define: $$ A\cdot p := p( A\cdot (x,y)^t)=p(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$$ where $t$ denotes transpose. The representations $V_m$ are, in a sense, all the irreducible representations of $G$ .
Theorem 1 For $0\leq m \leq p-1$ , the representations $V_m$ are irreducible representations of dimension $m+1$ over $\F$ . Furthermore, up to isomorphism, there are no other irreducible representations of $G$ over $\F$ .
- 1
- Charles B. Thomas, Representations of Finite and Lie Groups, Imperial College Press, London.
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"irreducible representations of the special linear group over  " is owned by alozano.
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Cross-references: isomorphism, dimension, transpose, structure, variables, independent, degree, homogeneous polynomials, vector space, proof, representations, irreducible, group, field, special linear group, prime
This is version 2 of irreducible representations of the special linear group over  , born on 2005-03-29, modified 2005-03-29.
Object id is 6914, canonical name is IrreducibleRepresentationsOfTheSpecialLinearGroupOverMathbbF_p.
Accessed 1724 times total.
Classification:
| AMS MSC: | 20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields) |
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Pending Errata and Addenda
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