Essential dimension of algebraic groups and integral representations of Weyl groups
Let \(G\) be a simple algebraic group over the complex numbers. The essential dimension \(\text{ed}(G)\) of \(G\) is the minimum dimension that a variety \(Y\) with free \(G\) action may have, if one requires the existence of a dominant equivariant map onto \(Y\) from an open subset of a linear \(G\) variety. The aim of this paper is to get better upper estimates on \(\text{ed}(G)\). To this end the author introduces for an integer lattice \(L\) with an action of the Weyl group \(W\) a variant \(\text{ed}_W(L)\) of essential dimension, formulated entirely in terms of \(W\)-lattices. If \(G\) is simple of adjoint type it is shown that \(\text{ed}(G)\leq\text{ed}_W(\mathbb Z\Phi)\), where \(\mathbb Z\Phi\) is the root lattice. So now it becomes a problem of bounding \(\text{ed}_W(\mathbb Z\Phi)\). Some general results are obtained and many special ones, also bounding some other \(\text{ed}_W(L)\). Here are a few general estimates, always with \(G\) adjoint of rank \(n\). If \(G\) is not of type \(\text{A}_1\) or \(\text{B}_n\), let \(\Phi_0\) consist of the short roots. Then \(\text{ed}_W(\mathbb Z\Phi)\leq | \Phi_0| -n-1\). And if \(G\) is of type \(\text{A}_1\) or \(\text{B}_n\), then \(\text{ed}_W(\mathbb Z\Phi)\leq 2n\). Further, for \(n\geq4\), \(\text{ed}(\text{PGL}_n)\leq \text{ed}_{S_n}(\mathbb Z\text{A}_{n-1})\leq n^2-3n+1\).
scientific article; zbMATH DE number 2155351