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#### On the irreducibility of irreducible characters of simple Lie algebras

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https://doi.org/10.1090/S0002-9947-2014-06080-7

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arXiv:1110.5300.pdf

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##### Citation

Rajan, C. S. (2014). On the irreducibility of irreducible characters of simple Lie
algebras.* Transactions of the American Mathematical Society,* *366*(12),
6443-6481. doi:10.1090/S0002-9947-2014-06080-7.

Cite as: https://hdl.handle.net/21.11116/0000-0004-DCA0-F

##### Abstract

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple

algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors.

For $SL(r)$ the irreducibility result is the following: let $\lambda=(a_1\geq

a_2\geq ... a_{r-1}\geq 0)$ be the highest weight of an irreducible rational representation $V_{\lambda}$ of $SL(r)$. Assume that the integers $a_1+r-1, ~a_2+r-2,..., a_{r-1}+1$ are relatively prime. Then the character $\chi_{\lambda}$ of $V_{\lambda}$ is strongly

irreducible in the following sense: for any natural number $d$, the function

$\chi_{\lambda}(g^d), ~g\in SL(r,\C)$ is irreducible in the ring of regular functions of $SL(r,\C)$.

algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors.

For $SL(r)$ the irreducibility result is the following: let $\lambda=(a_1\geq

a_2\geq ... a_{r-1}\geq 0)$ be the highest weight of an irreducible rational representation $V_{\lambda}$ of $SL(r)$. Assume that the integers $a_1+r-1, ~a_2+r-2,..., a_{r-1}+1$ are relatively prime. Then the character $\chi_{\lambda}$ of $V_{\lambda}$ is strongly

irreducible in the following sense: for any natural number $d$, the function

$\chi_{\lambda}(g^d), ~g\in SL(r,\C)$ is irreducible in the ring of regular functions of $SL(r,\C)$.