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# GL-representations, symmetric polynomials, and geometry

One of the many applications of symmetric polynomials is to representation theory, and in this post I want to begin sketching out how.

Symmetric polynomials and the ring $\Lambda$ are involved in the representation theory of the symmetric group $S_n$, and the general linear group $GL_n$, in related ways. The precise relationship between the representation theory of these two groups is spelled out in the Schur-Weyl Duality theorem, as well as in explicit constructions of representations of both groups.

I’m mainly interested in the Schur functors, which are representations of $GL_n$, so I’ll be focusing on those.

$GL$ representations and symmetric polynomials

Let’s first see how $GL_n$ representations are related to symmetric functions. Suppose

$\displaystyle{\rho : GL_n \to GL_m}$

is a continuous representation, and let $\chi$ be the trace. (I’ll only be interested in continuous, and soon enough, smooth representations.)

If $g \in GL_n$ is a diagonal matrix, then certainly $\chi(g)$ is a symmetric function of the entries of $g$, since these can be rearranged by conjugation. More generally, if $g$ is diagonalizable, then by conjugation-invariance again, $\chi(g)$ is a symmetric function of the eigenvalues of $g$. But diagonalizable matrices are dense in $GL_n$, so the same holds for all elements $g$. (This is why continuity is important.)

Next, let’s restrict to algebraic representations. Then the function $\chi(g)$ must be a (symmetric) polynomial in the eigenvalues! In fact, we’ll see later that it’s enough to restrict to smooth representations — all of them are rational, and most are polynomial.

In any case, the ring operations on symmetric polynomials are obviously compatible with direct sums and tensor products of representations, so we have a map

$\displaystyle{R(GL_n) \to \Lambda_n,}$

where $R(GL_n)$ is the representation ring and $\Lambda_n = k[x_1, \ldots, x_n]^{S_n}$. Note that the latter is a polynomial ring on $e_1, \ldots, e_n$ with no relations (it’s isomorphic to $\Lambda / (e_{n+1}, e_{n+2}, \ldots)$.)

A few important geometric facts about smooth representations of $GL_n$:

Fact. Every smooth representation of $GL_n$ is semisimple, that is, splits as a direct sum of irreducible representations.

Fact. A smooth representation of $GL_n$ is determined up to isomorphism by its character.

The former follows by relating $GL_n$ to the compact subgroup of unitary matrices $U_n$, where semisimplicity is automatic. The latter follows from the theory of Lie algebras, rephrasing characters and smooth representations in terms of weights of Lie algebra representations. I’ll come back to these later.

Also, this tells us that the map $R(GL_n) \to \Lambda_n$ is injective.

Some Examples

Let’s see some examples of $GL$ representations. At this point, I’ll write things in terms of $GL(V)$ for some vector space $V$, rather than $GL_n$: this has the advantage of illustrating how all of these constructions will be functorial in $V$.

First, there’s the tautological representation $V$ itself. The trace is just “the” trace,

$\displaystyle{\chi(g) = x_1 + \cdots + x_n,}$

which we have variously referred to as the polynomial $m_1, e_1, h_1, s_1$ and $p_1$.

Next, we have the exterior powers $\bigwedge^k V$. Recall that if $v_1, \ldots, v_n$ is a basis for $V$, then a basis for the exterior power is given by

$\displaystyle{v_{i_1} \wedge \cdots \wedge v_{i_k},}$

running over all choices of size-$k$ subsets $I = \{i_1, \ldots, i_k\}$. If $g$ is diagonal with entries $x_1, \ldots, x_n$, then each of the above basis elements is already an eigenvector, with eigenvalue given by the monomial $x^I$. Hence, the trace is

$\displaystyle{\chi(g) = \sum_{|I|=k} x^I = e_k,}$

the elementary symmetric function! In particular, this shows that the ring map $R(GL_n) \to \Lambda_n$ is surjective, hence an isomorphism. That is a bit of a surprise.

The other commonly-used $GL$ representations are the symmetric powers $S^k(V)$. Here, a basis is given by all symmetric products

$\displaystyle{v_{i_1} \cdots v_{i_k},}$

running over all choices of $k$ indices with repetition. By similar reasoning to the above, the trace is

$\displaystyle{\chi(g) = \sum \text{monomials of total degree } k = h_k,}$

the homogeneous symmetric polynomial.

The Schur Functors

Here’s an example of a representation that doesn’t fall into either of the above categories. Let

$\displaystyle{W = \text{Span}\big\{v_1 \otimes (v_2 \wedge v_3) : v_1 \in \text{Span}(v_2,v_3) \big\} \subset V \otimes \bigwedge\ ^2 V.}$

In other words, thinking of a pure wedge $v_2 \wedge v_3$ as a 2-plane, $W$ is spanned by “flag tensors” of lines contained in 2-planes.

This is a valid (nonzero) subrepresentation of $V \otimes \bigwedge^2 V$, since the definition is $GL(V)$-invariant. It’s also clearly functorial in $V$, so it is a “natural” representation. That said, none of the following questions have obvious answers:

• Is $W$ proper, or is it the entirety of $V \otimes \bigwedge^2 V$?
• how can we find a basis for $W$?
• what is the trace of $W$?

Proposition. The representation $W$ is proper.

Proof. Suppose $v_1 \otimes v_2 \wedge v_3$ is a generator, and $v_1 = a v_2 + b v_3$, and without loss of generality suppose $a \ne 0$. Then, by various linearity properties,

$\begin{array}{rl} v_1 \otimes v_2 \wedge v_3 & = v_1 \otimes (\tfrac{1}{a}v_1 - \tfrac{b}{a} v_3) \wedge v_3 \\ &= \tfrac{1}{a}v_1 \otimes v_1 \wedge v_3 \\ &= v_2 \otimes v_1 \wedge v_3 \ +\ v_3 \otimes v_1 \wedge \tfrac{b}{a} v_3 \\ &= v_2 \otimes v_1 \wedge v_3 \ +\ v_3 \otimes v_1 \wedge (\tfrac{1}{a} v_1 - v_2) \\ &= v_2 \otimes v_1 \wedge v_3 \ +\ v_3 \otimes v_1 \wedge v_2. \end{array}$

Of course, the same thing holds if instead $b \ne 0$, and therefore holds on all of $W$. But, of course, the relation

$v_1 \otimes v_2 \wedge v_3 = v_2 \otimes v_1 \wedge v_3 \ +\ v_3 \otimes v_1 \wedge v_2$

does not hold on all of $V \otimes \bigwedge^2 V$, where a basis is given by all choices of $v_i \otimes v_j \wedge v_k$.

So this is something different. It’s called the $(2,1)$ Schur functor, or $\mathbb{S}^{2,1}(V)$ for short, and the relation above is called an exchange relation, since it only involved permuting the vectors $v_i$.

Similar constructions for arbitrary “flag tensors” will give rise to a range of other new representations, such as

$\mathbb{S}^{5,4,1}(V) \subset V \otimes \bigwedge\ ^4(V) \otimes \bigwedge\ ^5(V).$

Of course, these are indexed by partitions, but so far only partitions with all distinct parts have shown up. Repeated parts will be slightly more subtle: if $\lambda = (3,3,1)$, for example, we’ll take the second symmetric power of the 3 part:

$\mathbb{S}^{3,3}(V) \subset V \otimes S^2(\bigwedge\ ^3(V)).$

These are all, it turns out, new.

A Geometric Perspective

My choice of terminology, “flag tensors”, was deliberate. What’s really going on here, invoking some algebraic geometry, is that the flag manifold $Fl(V)$ has a canonical Plücker embedding

$\displaystyle{Fl(V) \to \prod_{k=1}^n Gr(k,V) \to \prod_{k=1}^n \mathbb{P}(\bigwedge^k (V)).}$

This embedding gives us a multihomogeneous coordinate ring, which is multigraded (from each of the factors in the product). Moreover, the multigraded components of this ring really are “flag tensors” in the form I described above. In other words:

Theorem. The Schur functors are the multigraded components of the coordinate ring of the (complete) flag variety in its (product) Plücker embedding.

That’s enough for now. Next post, I’ll talk further about the actual construction of the Schur functors.

(Note: I’m being slightly non-canonical at the end here, since I described the Schur functors above as subrepresentations, but the multigraded components of the coordinate ring are quotient representations, since the ring is a quotient of the multigraded coordinate ring of the ambient projective space. To fix this, I should have used the dual flag variety $Fl(V^*)$.)