arXiv subjects

Alexander Lang

Publications and source records attributed to Alexander Lang.

3 recordsLinked to original sources

arXiv2024

Canonical structure constants for simple Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$. In the 1950s Chevalley showed that $\mathfrak{g}$ admits particular bases, now called ``Chevalley bases'', for which the corresponding structure constants are integers. Such bases are not unique but, using Lusztig's theory of canonical bases, one can single out a ``canonical'' Chevalley basis which is unique up to a global sign. In this paper, we give explicit formulae for the structure constants with respect to such a basis.

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arXiv2017

An Enumeration of the Supercharacter Theories of $C_p \times C_2 \times C_2$ for Prime $p$

The supercharacter theories of $C_p \times C_2 \times C_2$ were classified in the language of Schur rings by Evdokimov, Kovács, and Ponomarenko in [EKP16]. It was shown that every nontrivial supercharacter theory of $C_p \times C_2 \times C_2$ can be constructed as a wedge product, a direct product, or is generated by automorphisms. We use this classification to give a precise count of the distinct supercharacter theories of $C_p\times C_2\times C_2$ and describe when a supercharacter theory can be constructed by more than one method. We also present an alternative proof of the classification using the language of supercharacter theories.

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arXiv2014

Supercharacter Theories and Semidirect Products

We describe the supercharacter theories of the semidirect product of H and K, $H\rtimes K$ in terms of the supercharacter theories of the direct product of H and K in the case when both H and K are Abelian groups. To do this we introduce the concept of a homomorphism of supercharacter theories. This provides a classification of the supercharacter theories of the dihedral groups of order 2m when m is odd using the known classification of the supercharacter theories of cyclic groups.

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