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Gelfand-Tsetlin basis for the representations of $\gl_n$

1. Finite dimensional representations of $\gl_n$

Let $e_{ij}$, $i,j=1,\dots,n$ denote the standard basis of the general linear Lie algebra $\gl_n$ over the field of complex numbers. The subalgebra $\gl_{n-1}$ is spanned by the basis elements $e_{ij}$ with $i,j=1,\dots,n-1$. Denote by $\h=\h_n$ the diagonal Cartan subalgebra in $\gl_n$. The elements $e_{11},\dots,e_{nn}$ form a basis of $\h$.

Finite-dimensional irreducible representations of $\gl_n$ are in a one-to-one correspondence with $n$-tuples of complex numbers $\la=(\la_1,\dots,\la_n)$ such that
$\la_i-\la_{i+1}\in\Z^+\quad\text{for }i=1,\dots,n-1.$Such an $n$-tuple $\la$ is called the highest weight of the corresponding representation which we shall denote by $L(\la)$. It contains unique, up to a multiple, nonzero vector $v^+$ (the highest vector) such that $e_{ii}v^+=\la_iv^+$ for $i=1,\dots,n$ and $e_{ij}v^+=0$ for $1\lle i < j \lle n$.

Properties of Wronskians

This is a note for the article Critical points of master functions and flag varieties. Most of the properties below are from the appendix of this article. Some are not written in the paper but used there. I did not write the well-known relations between differential equations and Wronskians, such as Abel’s identity.

Here we assume all functions in this page are functions of $x$ with sufficiently many derivatives.

Define the Wronskian of functions $g_1,\dots,g_n$ by $W(g_1,\dots,g_n)=\det(g_i^{(j-1)})_{i,j=1}^n$. We follow the convention that for $n=0$ the corresponding Wronskian is equal to $1$. We also write $W_n(g_1,\dots,g_n)$ for $W(g_1,\dots,g_n)$ in order to stress the order of the Wronskian.

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