Polynomials and Representations XXIX

Characters

Definition. The character of a continuous G-module V is defined as:

\chi_V : G\to\mathbb{C}, \quad g \mapsto \text{tr}(g: V\to V).

This is a continuous map since it is an example of a matrix coefficient.

Clearly \chi_V(hgh^{-1}) = \chi_V(g) for any g,h\in G. The following are quite easy to show:

\displaystyle \begin{aligned}\chi_{V\oplus W}(g) &= \chi_V(g) + \chi_W(g); \\ \chi_{V\otimes_{\mathbb C} W}(g) &= \chi_V(g) \chi_W(g); \\ W\subseteq V\implies \chi_V(g) &= \chi_W(g) + \chi_{V/W}(g); \\ \chi_{V^\vee}(g) &= \chi_V(g^{-1}) = \overline{\chi_V(g)}.\end{aligned}

The last equality, that \chi_V(g^{-1}) is the complex conjugate of \chi_V(g), follows from the following:

Lemma 1. For g\in G, the linear map g:V\to V has all eigenvalues on the unit circle T.

Proof

Suppose the eigenvalues are \lambda_1, \ldots, \lambda_N with the right multiplicity. Then g^m has trace \chi_V(g^m)= \sum_i \lambda_i^m. But \chi_V : G\to \mathbb{C} is continuous; since G is compact the image is bounded. It remains to show: if \sum_i \lambda_i^m is bounded for all integers m we must have all |\lambda_i| = 1.

Suppose |\lambda_1| \ne 1; replacing g by its inverse, we assume |\lambda_1| > 1 and |\lambda_1| \ge |\lambda_2| \ge \ldots. If |\lambda_1| > |\lambda_2|, |\lambda_1|^m increases much faster than the other |\lambda_i|^m and we get a contradiction. On the other hand, if |\lambda_1| = \ldots = |\lambda_k|, write \lambda_j = Re^{i\theta_j} for 1\le j \le k where R>1 and \theta_j are real. Consider the continuous homomorphism

\phi:\mathbb{Z} \mapsto (\mathbb{C}^*)^k, \ m \mapsto (e^{im\theta_1}, \ldots, e^{im\theta_k}).

Its image lies in the torus group T_k which is compact. Hence, there are arbitrarily large m such that \text{Re}(e^{im\theta_1}), \ldots, \text{Re}(e^{im\theta_k}) > \frac 1 2 (see exercise). So the real part of \lambda_1^m + \ldots + \lambda_k^m exceeds \frac 1 2 R^m and increases much faster than the remaining |\lambda_i|^m. ♦

Exercise

Prove that if f:\mathbb{Z} \to G is a continuous homomorphism to a compact Hausdorff group, then f^{-1}(U) is infinite for any open neighbourhood U of e.

Lemma 2. For g\in G, the linear map g:V\to V is diagonalizable.

Proof

Write g:V\to V in block Jordan form. Suppose the matrix is not diagonal, say entry (1, 2) is 1 with corresponding eigenvalue \lambda. The matrix for g^n : V\to V has entry (1, 2) equal to n \lambda^{n-1} which is unbounded since |\lambda| = 1. However, G is compact so taking entry (1, 2) of the representation G\to GL(V) has bounded image, which is a contradiction. ♦

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Orthogonality of Characters

The space of G-invariant elements of a G-module V is denoted by:

V^G := \{v\in V: gv = v \, \forall\, g\in G\}.

We have:

Proposition. The dimension of V^G is:

\displaystyle \int_{x\in G} \chi_V(x) dx.

Proof

Take \pi: V\to V via v\mapsto \int_{x\in G} xv\cdot dx.

Left-invariance then implies that the image of π lies in V^G. Also, if v\in V^G then \pi(v) = v. Putting these two facts together gives us \pi^2 = \pi so  π is the projection map onto V^G. Hence:

\displaystyle \dim V^G = \text{tr}(\pi) = \int_{x\in G} \text{tr}(x:V\to V)dx = \int_{x\in G} \chi_V(x)\cdot dx.

as desired. ♦

In particular, replacing V by \text{Hom}_{\mathbb C}(V, W), note that:

\text{Hom}_{\mathbb C}(V, W)^G = \text{Hom}_{\mathbb C[G]}(V, W).

So the dimension of this space is the integral of:

\chi_{\text{Hom}(V, W)}(g) = \chi_{V^\vee \otimes W}(g) = \overline{\chi_V(g)} \chi_W(g).

In other words:

\dim \text{Hom}_{\mathbb{C}[G]}(V, W) = \int_{x\in G} \overline{\chi_V(x)} \chi_W(x)\cdot dx

which we denote by \left<V, W\right>. Schur’s lemma then gives us:

Orthonormality of Irreducible Characters.

If V, W are irreps of G, then

\left< V, W\right> = \int_{x\in G}\overline{\chi_V(x)} \chi_W(x)\cdot dx= \begin{cases} 1, \ &\text{if } V\cong W, \\ 0 \ &\text{else.}\end{cases}

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Abelian Case

Suppose now G is compact Hausdorff and abelian.

Theorem. Each irreducible representation V of G is 1-dimensional.

Proof

First we show that every g\in G acts as a scalar. By lemma 2 above, V decomposes as a direct sum of eigenspaces W_\lambda for g:V\to V. Each W_\lambda is G-invariant since:

g' \in G, v\in W_\lambda \implies g(g'v) = g'(gv) = g'(\lambda v) = \lambda (g'v) \implies g'v \in W_\lambda.

Since V is irreducible, we have W_\lambda = V so g acts as a scalar.

Since every element of G acts as a scalar, every vector subspace is G-invariant; so dim(V) = 1. ♦

Hence by lemma 1, every irrep of G is a continuous homomorphism G\to T, where T\subset \mathbb{C}^* is the unit circle. The following special case is of interest.

Definition. The n-torus is the topological group T^n \cong (\mathbb{R}/\mathbb{Z})^n.

[In fact, any compact Hausdorff abelian G must be of the form (finite group) \times T^n but we won’t be needing this fact.]

For i = 1,…,n, let \chi_i : T^n \to T be the homomorphism taken by projecting to the i-th coordinate. More generally, for any \mathbf m := (m_1, \ldots, m_n) \in \mathbb{Z}^n, we define:

\chi^{\mathbf m} := \chi_1^{m_1} \cdot \ldots \cdot \chi_n^{m_n} : T^n \to T.

Thus it takes \mathbf v =(v_1, \ldots, v_n) to v_1^{m_1} v_2^{m_2} \ldots v_n^{m_n}.

Theorem. Any irrep of T^n is of the form \chi^{\mathbf m} for some \mathbf m \in \mathbb{Z}^n.

Proof

Suppose n=1: for a continuous \chi:T\to T write T as \mathbb{R} / \mathbb{Z} and compose \mathbb{R} \to \mathbb{R}/\mathbb{Z} \stackrel\chi\to \mathbb{R}/\mathbb{Z}. The subgroup \mathbb{Z} lies in the kernel; since the kernel is a closed subgroup it is either \mathbb{R} or \frac 1 m \mathbb{Z} for some m>0. So \chi is of the form v\mapsto v^m for some m\in\mathbb{Z}.

For n>1 consider the homomorphism \phi_i : T \to T^n for = 1,…,n, by taking v\mapsto (1,\ldots, 1,v,1\ldots,1), where the i-th component is v. By the previous case, \chi\circ\phi_i : T\to T is of the form v\mapsto v^{m_i} for some m_i\in\mathbb{Z}. Since \mathbf v = \prod_i \phi_i\chi_i(\mathbf v) for all \mathbf v\in T^n, we get:

\chi(\mathbf v) = \prod_i \chi \phi_i\chi_i(\mathbf v) = \prod_i \chi_i(\mathbf v)^{m_i} = \chi^{\mathbf m}(\mathbf v). ♦

Corollary. A character of T^n is a finite linear combination of \chi^{\mathbf m} whose coefficients are positive integers.

Replacing \chi_i with variables x_i, this gives us a polynomial in x_1, \ldots, x_n, x_1^{-1},\ldots, x_n^{-1}, also known as a Laurent polynomial in x_1, \ldots, x_n. Hence we have a correspondence:

correspondence_torus_and_laurent_poly

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