arXiv subjects

Mikaël Cavallin

Publications and source records attributed to Mikaël Cavallin.

4 recordsLinked to original sources

arXiv2017

A new family of irreducible subgroups of the orthogonal algebraic groups

Let $n\geq 3,$ and let $Y$ be a simply connected, simple algebraic group of type $D_{n+1}$ over an algebraically closed field $K.$ Also let $X$ be the subgroup of type $B_n$ of $Y,$ embedded in the usual way. In this paper, we correct an error in a proof of a theorem of Seitz, resulting in the discovery of a new family of triples $(X,Y,V),$ where $V$ denotes a finite-dimensional, irreducible, rational $KY$-module, on which $X$ acts irreducibly. We go on to investigate the impact of the existence of the new examples on the classification of the maximal closed connected subgroups of the classical algebraic groups.

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arXiv2017

Structure of certain Weyl modules for the Spin groups

Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and let $W$ be a finite-dimensional $K$-space of dimension greater than or equal to $5.$ In this paper, we give the structure of certain Weyl modules for $G=\mbox{Spin}(W)$ in the case where $p\neq 2,$ as well as the dimension of the corresponding irreducible, finite-dimensional, rational $KG$-modules. In addition, we determine the composition factors of the restriction of certain irreducible, finite-dimensional, rational $K\mbox{SL}(W)$-modules to $\mbox{SO}(W).$

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arXiv2016

Restriction of irreducible modules for $\mbox{Spin}_{2n+1}(K)$ to $\mbox{Spin}_{2n}(K)$

Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and let $Y=\mbox{Spin}_{2n+1}(K)$ $(n\geqslant 3)$ be a simply connected simple algebraic group of type $B_n$ over $K.$ Also let $X$ be the subgroup of type $D_n,$ embedded in $Y$ in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for $Y.$ In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational $KY$-modules on which $X$ acts with exactly two composition factors.

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arXiv2016

An algorithm for computing weight multiplicities in irreducible modules for complex semisimple Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ having rank $l$ and let $V=L(λ)$ be an irreducible finite-dimensional $\mathfrak{g}$-module having highest weight $λ.$ Computations of weight multiplicities in $V,$ usually based on Freudenthal's formula, are in general difficult to carry out in large ranks or for $λ$ with large coefficients (in terms of the fundamental weights). In this paper, we first show that in some situations, these coefficients can be "lowered" in order to simplify the calculations. We then investigate how this can be used to improve the aforementioned formula of Freudenthal, leading to a more efficient version of the latter in terms of complexity as well as a way of dealing with certain computations in unbounded ranks. We conclude by illustrating the last assertion with a concrete example.

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