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edited Apr 25, 2013 at 21:42

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I'm struggling to see what the actual question is but here is an example of a non-trivial use of geometry to prove a simple statement in representation theory; moreover, it is the only known way to obtain the result (apologies if this is not what you're after): the $n!$ conjecture states that the dimension of a certain bigraded $S_{n}$-module is $n!$ (in fact, something stronger is true, the bigraded module is the left regular representation). This statement is equivalent to a certain morphism being Gorenstein and Cohen-Macaulay, namely the morphism $\rho: X_{n}\to H_{n}$, where $H_n$ is the Hilbert scheme of n points in $\mathbb{C}^2$ and $X_n$ is the isospectral Hilbert scheme. Mark Haiman gave a proof of the geometric statement in 2000 (math.AG/0010246).

The bigraded module (call it $D_{\mu}$) considered here is the span of partial derivatives (with respect to $x$'s and $y$'s) of a bihomogeneous polynomial $\Delta_{\mu}(x_1,\ldots,x_n;y_1,\ldots,y_n)$, where $\mu$ is a partition of $n$. Furthermore, the 'bigraded multiplicity' of the simple $S_n$-module $V^{\lambda}$ in $D_{\mu}$ gives the coefficients of the Macdonald-Kostka polynomials, thereby proving their positivity (Macdonald's conjecture).

Again, I'm not sure this is an example of 'geometric representation theory' as most people see it, but it's a nice example of using geometry to solve a 'classical' representation theoretic problem. (I suppose my struggle to see the question is more to do with understanding what comes under 'geometric representation theory')

I'm struggling to see what the actual question is but here is an example of a non-trivial use of geometry to prove a simple statement in representation theory; moreover, it is the only known way to obtain the result (apologies if this is not what you're after): the $n!$ conjecture states that the dimension of a certain bigraded $S_{n}$-module is $n!$ (in fact, something stronger is true, the bigraded module is the left regular representation). This statement is equivalent to a certain morphism being Gorenstein and Cohen-Macaulay, namely the morphism $\rho: X_{n}\to H_{n}$, where $H_n$ is the Hilbert scheme of n points in $\mathbb{C}^2$ and $X_n$ is the isospectral Hilbert scheme. Mark Haiman gave a proof of the geometric statement in 2000 (math.AG/0010246).

The bigraded module (call it $D_{\mu}$) considered here is the span of partial derivatives (with respect to $x$'s and $y$'s) of a bihomogeneous polynomial $\Delta_{\mu}(x_1,\ldots,x_n;y_1,\ldots,y_n)$, where $\mu$ is a partition of $n$. Furthermore, the 'bigraded multiplicity' of the simple $S_n$-module $V^{\lambda}$ in $D_{\mu}$ gives the coefficients of the Macdonald-Kostka polynomials, thereby proving their positivity (Macdonald's conjecture).

Again, I'm not sure this is an example of 'geometric representation theory' as most people see it, but it's a nice example of using geometry to solve a 'classical' representation theoretic problem.

I'm struggling to see what the actual question is but here is an example of a non-trivial use of geometry to prove a simple statement in representation theory; moreover, it is the only known way to obtain the result (apologies if this is not what you're after): the $n!$ conjecture states that the dimension of a certain bigraded $S_{n}$-module is $n!$ (in fact, something stronger is true, the bigraded module is the left regular representation). This statement is equivalent to a certain morphism being Gorenstein and Cohen-Macaulay, namely the morphism $\rho: X_{n}\to H_{n}$, where $H_n$ is the Hilbert scheme of n points in $\mathbb{C}^2$ and $X_n$ is the isospectral Hilbert scheme. Mark Haiman gave a proof of the geometric statement in 2000 (math.AG/0010246).

The bigraded module (call it $D_{\mu}$) considered here is the span of partial derivatives (with respect to $x$'s and $y$'s) of a bihomogeneous polynomial $\Delta_{\mu}(x_1,\ldots,x_n;y_1,\ldots,y_n)$, where $\mu$ is a partition of $n$. Furthermore, the 'bigraded multiplicity' of the simple $S_n$-module $V^{\lambda}$ in $D_{\mu}$ gives the coefficients of the Macdonald-Kostka polynomials, thereby proving their positivity (Macdonald's conjecture).

Again, I'm not sure this is an example of 'geometric representation theory' as most people see it, but it's a nice example of using geometry to solve a 'classical' representation theoretic problem. (I suppose my struggle to see the question is more to do with understanding what comes under 'geometric representation theory')

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created Apr 25, 2013 at 21:13

1.2k
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15

I'm struggling to see what the actual question is but here is an example of a non-trivial use of geometry to prove a simple statement in representation theory; moreover, it is the only known way to obtain the result (apologies if this is not what you're after): the $n!$ conjecture states that the dimension of a certain bigraded $S_{n}$-module is $n!$ (in fact, something stronger is true, the bigraded module is the left regular representation). This statement is equivalent to a certain morphism being Gorenstein and Cohen-Macaulay, namely the morphism $\rho: X_{n}\to H_{n}$, where $H_n$ is the Hilbert scheme of n points in $\mathbb{C}^2$ and $X_n$ is the isospectral Hilbert scheme. Mark Haiman gave a proof of the geometric statement in 2000 (math.AG/0010246).

The bigraded module (call it $D_{\mu}$) considered here is the span of partial derivatives (with respect to $x$'s and $y$'s) of a bihomogeneous polynomial $\Delta_{\mu}(x_1,\ldots,x_n;y_1,\ldots,y_n)$, where $\mu$ is a partition of $n$. Furthermore, the 'bigraded multiplicity' of the simple $S_n$-module $V^{\lambda}$ in $D_{\mu}$ gives the coefficients of the Macdonald-Kostka polynomials, thereby proving their positivity (Macdonald's conjecture).

Again, I'm not sure this is an example of 'geometric representation theory' as most people see it, but it's a nice example of using geometry to solve a 'classical' representation theoretic problem.