Relation between two geometrically defined bases in representations of $GL_n$

Details Relation between two geometrically defined bases in representations of $GL_n$ Relation between two geometrically defined bases in representations of $GL_n$

2004-11-11

arxiv.org/abs/math/0411252v2

Mathematics

Representation Theory

Scientific paper

Let $V$ be an irreducible representation of group $GL_n({\mathbb C})$, which appears as a submodule in $({\mathbb C}^n)^{\otimes d}$, where ${\mathbb C}^n$ is the tautological $n$-dimensional representation of $GL_n$, and $d$ is a non-negative integer. On the one hand, following refs [Gi] and [BG] one can produce a basis in $V$ using irreducible components of Sringer fibers over a nilpotent matrix in ${\mathfrak {gl}}_d$, whose Jordan blocks correspond to the highest weight of $V$. On the other hand, one can produce a basis in $V$ by Mirkovi\'c-Vilonen cycles, a construction that works for an arbitrary reductive group $G$. In this note we prove that the resulting to bases coincide.

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