Polynomials and Representations XXXI

K-Representations and G-Representations

As mentioned at the end of the previous article, we shall attempt to construct analytic representations of G = GL_n\mathbb{C} from continuous representations of K = U_n.

Let \rho : K \to GL(V). Consider \rho|_S, where S is the group of diagonal matrices in K so

S = \left\{\begin{pmatrix} e^{i\theta_1} & 0 & \ldots & 0 \\ 0 & e^{i\theta_2} & \ldots & 0\\ \vdots & \vdots & \ddots &\vdots\\ 0 & 0 & \ldots & e^{i\theta_n}\end{pmatrix} : \theta_1, \ldots, \theta_n \in \mathbb{R}\right\} \cong (\mathbb R/\mathbb Z)^n

as a topological group. From our study of representations of the n-torus, we know that \rho|_S is a direct sum of 1-dimensional irreps of the form:

\rho_{\mathbf a} = \rho_{a_1, \ldots, a_n} : S \to \mathbb C^*, \quad \begin{pmatrix}x_1 & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & x_n \end{pmatrix} \mapsto x_1^{a_1} \ldots x_n^{a_n},

where \mathbf a = (a_1, \ldots, a_n) \in \mathbb Z^n. E.g. if \mathbf a = \mathbf 0 then the representation is trivial; if \mathbf a = (1,\ldots, 1), the representation is obtained by taking the determinant.

Hence, the character of \rho|_S is a Laurent polynomial with non-negative integer coefficients, i.e.

\chi(\rho|_S) \in \mathbb{Z}[x_1, \ldots, x_n, x_1^{-1}, \ldots, x_n^{-1}].

Definition. For a continuous finite-dimensional representation \rho :K \to GL (V), we will write \psi_V for \chi(\rho|_S) expressed as a Laurent polynomial in x_1, \ldots x_n.

The same holds for a complex analytic representation of G.

Examples

  1. If \rho: G \to GL (\mathbb C^n) is the identity map, its Laurent polynomial is x_1 + x_2 + \ldots + x_n.
  2. For \det : G\to \mathbb C^*, its Laurent polynomial is x_1 x_2 \ldots x_n.

The following are clear for any continuous representations VW of K.

\begin{aligned} \psi_{V\oplus W} &= \psi_V + \psi_W \\ \psi_{V\otimes W} &= \psi_V \cdot \psi_W\\ \psi_{V} &= \psi_W + \psi_{V/W} \text{ if } W\subseteq V \\ \psi_{V^\vee}(x_1, \ldots, x_n) &= \psi_V(x_1^{-1}, \ldots, x_n^{-1}).\end{aligned}

In summary, so far we have the following:

representations_correspondences

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Main Examples: Sym and Alt

For any vector space V, the group S_d acts on V^{\otimes d} by permuting the components. We denote:

\begin{aligned} \text{Sym}^d V &:= \{ v \in V^{\otimes d} : w(v) = v \text{ for each } w\in S_d\},\\ \text{Alt}^d V &:= \{ v \in V^{\otimes d} : w(v) = \chi(w)v \text{ for each } w\in S_d\}\end{aligned}

where \chi(w) is the sign of w. Let e_1, \ldots, e_n be a fixed basis of V. The case d=2 is quite easy to describe for the above spaces, for we can just take the following bases:

  • \text{Sym}^2 V : \{ e_i \otimes e_j + e_j \otimes e_i : 1 \le i \le j \le n\};
  • \text{Alt}^2 V : \{ e_i \otimes e_j - e_j \otimes e_i : 1 \le i < j \le n\}.

Denote these two types of elements by e_i e_j and e_i \wedge e_j respectively, so that e_i e_j = e_j e_i and e_i \wedge e_j = -e_j \wedge e_i. Note that V\otimes V \cong \text{Sym}^2 V \oplus\text{Alt}^2 V; this is not true for higher values of d.

Similarly, in general, we can pick the following as bases:

  • \text{Sym}^d V : \{ e_{i_1} \ldots e_{i_d} : 1 \le i_1 \le \ldots \le i_d \le n\};
  • \text{Alt}^d V : \{ e_{i_1} \wedge \ldots \wedge e_{i_d}: 1 \le i_1 < \ldots < i_d \le n\}.

where the components commute in Sym and anticommute in Alt (e.g. e_1 \wedge e_2 \wedge e_4 = - e_2 \wedge e_1 \wedge e_4 = e_2 \wedge e_4 \wedge e_1).

Now suppose V = \mathbb{C}^n and G = GL_n\mathbb{C} acts on it; G and S_d act on V^{\otimes d} in the following manner:

\begin{aligned} g\in G &\implies g(v_1 \otimes \ldots \otimes v_d) = g(v_1) \otimes \ldots \otimes g(v_d), \\ w\in S_d &\implies w(v_1 \otimes \ldots \otimes v_d) = v_{w^{-1}(1)} \otimes \ldots \otimes v_{w^{-1}(d)}.\end{aligned}

These two actions commute, from which one easily shows that \text{Sym}^d V and \text{Alt}^d V are G-invariant subspaces of V^{\otimes d}.

Example

Suppose we have g = \begin{pmatrix} 2 & -1\\ 1 & 3\end{pmatrix} acting on \text{Sym}^3 \mathbb{C}^2. This takes:

e_1^2 e_2 \mapsto (2e_1 - e_2)^2 (e_1 + 3e_2) = 4e_1^3 + 8e_1^2 e_2 - 11e_1 e_2^2 + 3e_2^3.

To compute the Laurent polynomials of these spaces, we let the diagonal matrix D(x_1, \ldots, x_n) act on them, giving:

\begin{aligned} e_{i_1} e_{i_2} \ldots e_{i_d} &\mapsto (x_{i_1} x_{i_2} \ldots x_{i_d}) e_{i_1} e_{i_2} \ldots e_{i_d},\\ e_{i_1} \wedge \ldots \wedge e_{i_d}& \mapsto (x_{i_1} x_{i_2}\ldots x_{i_d}) e_{i_1} \wedge \ldots \wedge e_{i_d}.\end{aligned}

Hence we have:

\displaystyle\begin{aligned} \psi_{\text{Sym}^d} &= \sum_{1\le i_1 \le \ldots \le i_d} x_{i_1} x_{i_2} \ldots x_{i_d} = h_d(x_1, \ldots, x_n),\\ \psi_{\text{Alt}^d} &= \sum_{1 \le i_1 < \ldots < i_d} x_{i_1} x_{i_2} \ldots x_{i_d} = e_d(x_1, \ldots, x_n).\end{aligned}

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Some Lemmas

Lemma 1. For a representation V of K, the Laurent polynomial \psi_V is symmetric.

Proof

For any w\in S_n, take the corresponding permutation matrix M\in K; we have:

\displaystyle M\cdot \begin{pmatrix} x_1 & \ldots & 0 \\ \vdots & \ddots & \vdots \\0 & \ldots & x_n \end{pmatrix}\cdot M^{-1} = \begin{pmatrix}x_{w(1)} & \ldots & 0 \\ \vdots & \ddots & \vdots \\0 & \ldots & x_{w(n)}\end{pmatrix}.

Thus \psi_V(x_1, \ldots, x_n) = \psi_V(x_{w(1)}, \ldots, x_{w(n)}) for any w\in S_n. ♦

Lemma 2. Given K-representations V, W, if \psi_{V} = \psi_{W}, then V\cong W.

Hence by the previous article, the same holds for analytic G-representations V, W.

Proof

Any M\in K, being unitary, is diagonalizable by a unitary matrix, i.e. there exists Q\in K such that QMQ^{-1} \in S. Hence the given condition implies:

\chi_{V}(M) = \chi_{V}(\overbrace{QMQ^{-1}}^{\in S})= \chi_{W}(QMQ^{-1}) = \chi_{W}(M)..

By character theory of compact groups, V \cong W as K-reps. ♦

Now for the final piece of the puzzle.

Lemma 3. Let f \in \mathbb{Z}[x_1, \ldots, x_n] be a symmetric polynomial. There are polynomial representations:

\rho_1 : G\to GL(V_1),\quad \rho_2 : G\to GL(V_2),

such that \psi_{V_1} - \psi_{V_2} = f.

Proof

Taking homogeneous parts, let us assume f \in \Lambda_n^{(d)} for some degree d\ge 0. Write f as a linear sum of elementary symmetric polynomials \{h_\lambda\}_{\lambda\vdash d, \lambda_1 \le n} with integer coefficients; separating terms we have f = g - h, where gh are both linear combinations of \{h_\lambda\} with non-negative integer coefficients. Hence, it suffices to show that h_\lambda = \psi_V for some polynomial representation G\to GL(V).

Since h_\lambda = h_{\lambda_1} \ldots h_{\lambda_l}, we can just pick:

V = \left(\text{Sym}^{\lambda_1} \mathbb{C}^n\right) \otimes \left(\text{Sym}^{\lambda_2} \mathbb{C}^n\right) \otimes \ldots \otimes \left(\text{Sym}^{\lambda_d} \mathbb{C}^n\right),

from the above. ♦

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Consequences

Immediately we have:

Corollary 1. For any symmetric Laurent polynomial f\in \mathbb{Z}[x_1^{\pm 1}, \ldots, x_n^{\pm 1}], there exist rational representations:

\rho_1 : G\to GL(V_1), \quad \rho_2 : G\to GL(V_2)

such that f = \psi_{V_1} - \psi_{V_2}.

Proof

Indeed, f = (x_1\ldots x_n)^{-M}g for some symmetric polynomial g in x_1, \ldots x_n. Applying lemma 3, we can find \rho_1, \rho_2 such that g = \psi_{V_1} - \psi_{V_2}; now we take \rho_1 \otimes \det^{-M} and \rho_2\otimes \det^{-M}. ♦

Finally we have:

Corollary 2. Any irreducible K-module V can be lifted to a rational irreducible G-module.

Proof

By corollary 1, \psi_V = \psi_{W} - \psi_{W'} where WW’ are rational G-modules. Then \psi_W = \psi_V + \psi_{W'} = \psi_{V\oplus W'} so W\cong V\oplus W' as K-modules by lemma 2 above. By corollary 3 here, V\subseteq W is a rational irreducible G-module. ♦

Summary of Results

Thus we obtain:

representations_correspondences_v2

Note the following.

  • When we consider virtual representations (recall that these are formal differences of two representations), this corresponds to all symmetric Laurent polynomials with integer coefficients.
  • Any rational G-representation is of the form \rho \otimes \det^m where \rho is a polynomial G-representation.

Moving Ahead

Our next task is to identify the irreducible rational G-modules V. Tensoring by some power of det, we may assume V is polynomial, so \psi_V(x_1, \ldots, x_n) is a symmetric polynomial in x_1, \ldots, x_n. Studying these will lead us back to Schur polynomials.

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