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We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original conjecture to all nilpotent orbits and also takes into account the action of the component group. The method is to use Shojis approach to the orthogonality formulas for Green functions to relate the symmetric algebra to a sum over Green functions. In the second part of the paper we give an explanation of the appearance of the Orlik-Solomon exponents using a result from rational Cherednik algebras.
In this paper we consider symmetric powers representation and exterior powers representation of finite groups, which generated by the representation which has finite dimension over the complex field. We calculate the multiplicity of irreducible compo
For given representation of finite groups on a finite dimension complex vector space, we can define exterior powers of representations. In 1973, Knutson found one of methods of calculating the character of exterior powers of representations with prop
A sequence of $S_n$-representations ${V_n}$ is said to be uniformly representation stable if the decomposition of $V_n = bigoplus_{mu} c_{mu,n} V(mu)_n$ into irreducible representations is independent of $n$ for each $mu$---that is, the multiplicitie
These myh lectures at the Park City conference in 1998.
Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and $n$. Let $phi$ be the canonical map from $Hom(V,W)$ to $Hom(wedge^t V,wedge^t W)$. We investigate the Zariski closure $X_t$ of the image $Y_t$ of $phi$. In the case $t=min(m