Polynomials and Representations XXXII

We attempt to identify the irreducible rational representations of G = GL_n\mathbb C. From the last article, we may tensor it with a suitable power of det and assume it is polynomial.

One key ingredient is the following rather ambiguous statement.

  • Peter-Weyl Principle: any irrep can be embedded inside the ring of functions on G.

To make sense of this statement, we need to define a group action on the latter. This article is rather general; throughout, G is allowed to be any topological group and all representations are assumed to be continuous and finite-dimensional.


C(G) as a Representation

Let C(G) be the ring of continuous functions G\to \mathbb{C}.

Definition. The action of G\times G on C(G) is defined as follows:

(x, y)\in G\times G, f \in C(G)\ \implies\ (x,y)\cdot f : G\to\mathbb{C}, g\mapsto f(x^{-1}gy).

One checks that (x',y')\cdot((x,y)\cdot f) = (x'x, y'y)\cdot f.

Some examples of functions in C(G) are the matrix coefficients; recall that these are functions of the form f : G \to \mathbb{C}, g\mapsto \alpha (\rho(v)) for some representation \rho : G\to GL(V), v\in V and \alpha \in V^\vee. A more economical way of writing this is f = \beta\circ \rho for some \beta \in \text{End}_{\mathbb C}(V)^\vee.

Proposition. Let \mathcal{O}(G) be the set of matrix coefficients. Then \mathcal{O}(G) is a (unital) \mathbb{C}-subalgebra of C(G).


To prove 1\in\mathcal{O}(G), let \rho :G \to \mathbb{C}^* be the trivial representation.

Closure under addition and multiplication: suppose \rho_i:G\to GL(V_i), \beta_i\in \text{End}(V_i)^\vee for i = 1,2; let f_i = \beta_i \circ \rho_i:G\to\mathbb{C} be the matrix coefficient. Taking:

\begin{aligned} \rho_1 \oplus \rho_2 : G \to GL(V_1 \oplus V_2), \ \ &\beta_1 \oplus \beta_2 \in \text{End}(V_1)^\vee \oplus \text{End}(V_2)^\vee \subseteq \text{End}(V_1 \oplus V_2)^\vee, \\ \rho_1\otimes \rho_2 : G\to GL(V_1 \otimes V_2), \ \ &\beta_1 \otimes \beta_2 \in \text{End}(V_1)^\vee \otimes \text{End}(V_2)^\vee \cong \text{End}(V_1 \otimes V_2)^\vee,\end{aligned}

we have f_1 + f_2 = (\beta_1 \oplus \beta_2) \circ (\rho_1 \oplus \rho_2) and f_1 f_2 = (\beta_1 \otimes \beta_2) \circ(\rho_1 \otimes \rho_2). ♦


Identifying Matrix Coefficients in C(G)

Next we have:

Lemma. For any f\in C(G), the following are equivalent.

  1. The span of all (g,g')\cdot f over all (g, g')\in G \times G is finite-dimensional.
  2. The span of all (g,1)\cdot f over all g\in G is finite-dimensional.
  3. The span of all (1,g)\cdot f over all g\in G is finite-dimensional.
  4. f is a matrix coefficient.


That 1 ⇒ 2, 3 is obvious. Let us now prove 4 ⇒  1. Suppose f = \beta\circ \rho for \rho : G\to GL(V) and \beta : \text{End}(V) \to \mathbb{C}. Then the function (x, y)f : G\to \mathbb{C} takes

g \mapsto f(x^{-1}gy) = \beta(\rho(x^{-1}) \rho(g) \rho(y)) = \beta'(\rho(g))

where \beta' : \text{End} (V) \to \mathbb{C} is a linear map which takes \phi \mapsto \beta(\rho (x^{-1})\phi\rho(y)). Hence the span of all (x,y)f lies in \{\beta' \circ \rho : \beta' \in \text{End}(V)^\vee\} and (1) follows since \text{End}(V)^\vee is finite-dimensional.

Finally, we prove 3 ⇒ 4; the case of 2 ⇒ 4 is similar. Let V be the space spanned by all (1, g)f, so V is a finite-dimensional subspace of C(G) which is (1\times G)-invariant. Let

  • \rho : G \to GL(V) be given by the action of (1\times G) on V;
  • \beta : \text{End}(V) \to \mathbb{C} be the map \alpha \mapsto (\alpha(f))(e_G).

Now tracing through \beta\circ \rho : G \to \mathbb{C} gives:

\overbrace{g}^{\in G}\ \mapsto \overbrace{(f_1\in V \mapsto (x\mapsto f_1(xg)))}^{\in GL(V)} \mapsto \left.(x\mapsto f(xg))\right|_{x=e} = f(g).

Thus f = \beta\circ \rho is a matrix coefficient. [Note: for 2 ⇒ 4, one needs to take the dual of V.]   ♦

Corollary. \mathcal{O}(G) is the sum of all finite-dimensional G\times G-subrepresentations of C(G).


Weak Peter-Weyl Theorem

Let V be a representation of G; the vector space End(V) becomes a G\times G-rep via:

(x,y) : (\alpha : V\to V) \mapsto (\rho_V(x)\circ \alpha \circ\rho_V(y^{-1}) : V\to V).

Another way of seeing this is: \text{End}(V) \cong V\otimes V^\vee naturally and the action of G\times G on the LHS is obtained from the RHS. Next, we obtain a linear map:

\text{End}(V)^\vee\to \mathcal{O}(G),\qquad \beta \mapsto (\beta \circ \rho_V : G\to \mathbb{C}). (*)

This is G\times G-equivariant:

  • Indeed (xy) takes \alpha\in \text{End}(V) to \rho_V(x) \alpha \rho_V(y^{-1}) so dually \beta \in \text{End}(V)^\vee is taken to the function (\alpha \in \text{End}(V)) \mapsto \beta(\rho_V(x^{-1})\alpha \rho_V(y)).
  • In \mathcal{O}(G), (xy) takes f:G\to\mathbb{C} to g\mapsto f(x^{-1}gy). Letting f = \beta\circ \rho_V it gets taken to g \mapsto \beta(\rho_V(x^{-1}) \rho_V(g) \rho_V(y)) so the following commutes:

\begin{matrix} \text{End}(V)^\vee &\longrightarrow & \mathcal{O}(G)\\ \downarrow & &\downarrow & \text{action of }(x,y)\\ \text{End}(V)^\vee &\longrightarrow &\mathcal{O}(G)\end{matrix}

warningWe have \text{End}(V) \cong V \otimes V^\vee so its dual V^\vee \otimes V is naturally isomorphic to End(V) as vector spaces and as G-reps, but not as G\times G-reps. This is because the isomorphism V^\vee \otimes V \cong V\otimes V^\vee requires a twist of the two components, which is not G\times G-equivariant.

Now suppose G is compact Hausdorff.

Lemma. Let V, W be irreducible representations of G. Then:

  • \text{End}(V) is an irreducible representation of G\times G;
  • \text{End}(V) \cong \text{End}(W) as G\times G-reps if and only if V\cong W as G-reps.


The character of End(V) is: \chi_{\text{End}(V)}(g_1, g_2) = \chi_{V}(g_1) \chi_{V^\vee}(g_2). Taking the inner product:

\begin{aligned}\int_{G\times G} \chi_{\text{End}(V)}(g_1, g_2) \chi_{\text{End}(W)}(g_1, g_2) d(g_1, g_2) &=\int_G \int_G \chi_{V}(g_1) \chi_{V^\vee}(g_2)  \chi_{W}(g_1) \chi_{W^\vee}(g_2) dg_1 \, dg_2\\ &= \left(\int_G \chi_{V}(g_1) \chi_{W}(g_1) dg_1\right) \left(\int_G \chi_{V^\vee}(g_2) \chi_{W^\vee}(g_2) dg_2\right)\\&= \begin{cases} 1, \ \ &\text{if } V\cong W; \\ 0,\ \ &\text{otherwise.}\end{cases}\end{aligned}

This proves both claims together. ♦

Taking the direct sum of (*) over all irreps V of G, we obtain:

\displaystyle\bigoplus_{V \text{irrep of } G} \text{End}(V)^\vee\to \mathcal{O}(G).

Weak Peter-Weyl Theorem. The above map is an isomorphism.


To show injectivity: the kernel of the map is a G\times G-submodule, so it is a direct sum of G\times G-irreps \text{End}(V)^\vee. On the other hand, \text{End}(V)^\vee \to \mathcal{O}(G) clearly cannot map to 0, since that would mean \rho_V = 0.

To show surjectivity: each f\in \mathcal{O}(G) is, by definition, of the form \beta \circ \rho for some \rho : G \to GL(V), \beta \in \text{End}(V)^\vee. Thus lies in the image. ♦

We remind the reader that this theorem only works when G is compact Hausdorff.

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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.


  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:



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}.


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}


Some Lemmas

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


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.


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.


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. ♦



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}.


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.


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:


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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Polynomials and Representations XXX

Representations of GLn and Un

Note: all representations of topological groups are assumed to be continuous and finite-dimensional.

Here, we will look at representations of the general linear group GL_n \mathbb{C}. We fix the following notations:

  • G denotes GL_n\mathbb{C} for some fixed n;
  • K is the subgroup U_n \subset GL_n(\mathbb{C}) of unitary matrices; this group is compact.

Now, a representation

\rho: G = GL_n\mathbb{C}\to GL_N\mathbb{C}

is said to be continuousanalyticrationalpolynomial if it is so in each entry of GL_N\mathbb{C}. For example:

  • G \to \mathbb{C}^*, \ g \mapsto \det(g)^{m} is rational for each integer n but polynomial only for non-negative values of m.
  • G \to \mathbb{C}^*,\ g \mapsto |\det(g)| is continuous but not analytic.

Note that the above definition makes sense for G since it is an open subset of \mathbb{C}^{n^2}, the space of all n\times n matrices.

Definition. For a coordinate-free variant, we say that the representation \rho : G = GL_n\mathbb{C} \to GL(V) is continuous / analytic / rational / polynomial if:

  • for any v\in V and \alpha \in V^\vee, the function G\to \mathbb{C}, g\mapsto \alpha(\rho(g)(v)) is so.

Our focus is in analytic representations of G. For K, we will only look at continuous representations since it does not have a natural analytic structure. It turns out there is a correspondence between these two classes of representations, so the former can be analyzed using our earlier tools.


Main Theorem

We begin with a basic result in complex analysis.

Theorem. Let f(z_1, \ldots, z_n) be an analytic function on \mathbb{C}^n. If f=0 on \mathbb{R}^n, then f=0.


When n=0, \mathbb{R}^n = \mathbb{C}^n has only one point so this is trivial. Suppose n>0 and f\ne 0. Treat f as a function in z_n and we get:

f(z_1, \ldots, z_n) = z_n^d g(z_1, \ldots, z_{n-1}) + z_n^{d+1} h(z_1, \ldots, z_n)

where g\ne 0 and h are analytic. Suppose f=0 on \mathbb{R}^n. Then f z_n^{-d} = g + z_n h is analytic and on \mathbb{R}^n - (\mathbb{R}^{n-1} \times \{0\}) it is zero; since this set is dense in \mathbb{R}^n, we have g + z_n h=0 on \mathbb{R}^n. Thus g=0 on \mathbb{R}^{n-1} \times \{0\}. By induction hypothesis, this gives g=0, a contradiction. ♦

From this, we obtain the following result, which is the main case of concern.

Main Lemma. If f : G\to\mathbb{C} is an analytic function and f|_K = 0, then f=0.


Let h : M_{n\times n}(\mathbb{C}) \cong \mathbb{C}^{n^2} \to \mathbb{C} be defined by h(A) := f(\exp(2\pi i A)), which is analytic. We have:

A \in M_{n\times n}(\mathbb{R})\implies \exp(2\pi i A) \in U_n= K

so h=0 when restricted to M_{n\times n}(\mathbb{R}). By the above theorem, h=0. Note that any diagonalizable matrix can be written as \exp(A) for some matrix A, so f(B) = 0 if B is diagonalizable. Since the set of diagonalizable matrices in GL_n\mathbb{C} is dense, f = 0. ♦



From the main lemma we get, in turn:

Corollary 1. Let V, W be finite-dimensional analytic representations of G. Then:

\text{Hom}_K(V, W) = \text{Hom}_G(V, W).


A linear map f:V\to W is G-equivariant if and only if for each g\in G\rho_W(g)\circ f = f\circ \rho_V(g). This is equivalent to:

\lambda \in \text{Hom}_\mathbb{C}(V, W)^\vee, g\in G \implies\lambda(\rho_W(g) \circ f - f\circ \rho_V(g))= 0.

Since f, \rho_V(g), \rho_W(g), \lambda are all analytic, by the main lemma, this condition remains equivalent when we replace G by K, in which case the condition says that f\in \text{Hom}_K(V, W). ♦

Corollary 2. Let V, W be finite-dimensional analytic G-modules. They are isomorphic as G-modules ⇔ they are isomorphic as K-modules.


Indeed, saying V\cong W (as G– or K-modules) just means there is an invertible f\in \text{Hom}_G(V, W) or \text{Hom}_K(V, W). Apply corollary 1. ♦

Corollary 3. Let V be a finite-dimensional analytic G-module. A subspace W\subseteq V is G-invariant ⇔ it is K-invariant. Hence V is irreducible as a G-module ⇔ it is irreducible as a K-module.


W is G-invariant if and only if for all g\in G, \rho_V(g)(W) \subseteq W; this is equivalent to:

  • if w\in W and \lambda \in V^\vee satisfies \lambda(W) =0, then the function G\to \mathbb{C}, g \mapsto \lambda(\rho_V(g)(w)) vanishes.

By corollary 1, this condition is equivalent when we replace G by K, but now the condition says \rho_V(g)(W)\subseteq W for all g\in K. ♦


Let V be a finite-dimensional analytic representation of G. Since K is compact, V as a K-module can be written as a direct sum of irreps:

V = W_1^{\oplus m_1} \oplus W_2^{\oplus m_2} \oplus \ldots \oplus W_r^{\oplus m_r},

where the W_i are pairwise non-isomorphic K-irreps. By corollary 3, each W_i is a G-irrep; the W_i are also pairwise non-isomorphic G-modules by corollary 2.


Given a K-module, can we necessarily lift it to an analytic G-module? We will explore this in the next article.


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Polynomials and Representations XXIX


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.


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. ♦


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.


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. ♦


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.


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}


Abelian Case

Suppose now G is compact Hausdorff and abelian.

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


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.


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:



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Polynomials and Representations XXVIII

Starting from this article, we will look at representations of G := GL_n\mathbb{C}. Now, GL_n\mathbb{C} itself is extremely complicated so we will only focus on representations of particular types. Generally, for any topological group G, we want:

\rho : G\to GL_N \mathbb{C}

to be a continuous homomorphism of groups.


Continuous Representations of Topological Groups

Let G be a topological group.

Our main objects of study are continuous representations \rho: G \to GL_N\mathbb{C}. These correspond to \mathbb{C}[G]-modules (also called G-modules) V of finite complex dimension, with an additional condition for continuity:

  • for any v\in V and \alpha \in V^\vee (dual), the map G\to \mathbb{C} given by g\mapsto \alpha(g(v)) is continuous.

[Note the above condition condition is independent of coordinates and does not require a choice of basis. When we replace G with GL_n\mathbb{C} later, we can also replace the word continuous with “analytic” to obtain a narrower class of representations, but let’s not get ahead of ourselves.]

A function G\to\mathbb{C} of the form g\mapsto \alpha(g(v)) as above is called a matrix coefficient. E.g. if we fix a basis for V which gives us G \to GL_N\mathbb{C}, picking entry (ij) in the matrix gives us a matrix coefficient G\to\mathbb{C}.

Henceforth, all representations of topological groups are assumed to be continuous and finite-dimensional.

Constructing New Representations

These constructions are identical to the case for representations of finite groups.

  • Given any complex vector space V, the trivial action takes gvv for any g\in G, v\in V; it is clearly continuous.
  • Given a G-module V, a vector space W\subseteq V is said to be Ginvariant if g\in G, w\in W \implies g(w) \in W. Clearly, the resulting G\to GL(W) is continuous.
    • We say that V is irreducible if V\ne 0 and it has no non-trivial G-invariant submodules. An irreducible representation V is often called irrep for short.
  • If W\subseteq V is G-invariant, then V/W is a G-module; clearly it is continuous.
  • If V is a G-module, the dual V^\vee is also continuous, where G acts via

G\times V^\vee \to V^\vee, \qquad (g, \alpha) \mapsto \alpha\circ g^{-1} \in V^\vee.

  • Given G-modules V, W, the action of G on V\oplus W and V \otimes_{\mathbb C} W are continuous as well.
  • Finally we have \text{Hom}_{\mathbb C}(V, W) \cong V^\vee \otimes_{\mathbb C} W naturally. Using the above constructions, we get a continuous G-representation for this space via:

g\in G, f:V\to W \mapsto \rho_W(g)\circ f \circ \rho_V(g^{-1}) : V\to W.

How do we know that the above constructions are all continuous?

Here’s one example. To show that V\otimes_{\mathbb{C}} W is a continuous representation, pick a basis e_i (resp. f_j) of V (resp. W) so that e_i \otimes f_j form a basis of V\otimes W. The matrix for g: V\otimes W \to V\otimes W is obtained from the matrices of g:V\to V and g:W\to W by multiplying entries; thus the map G\to \mathbb{C} taking g to a fixed entry of the matrix for g:V\otimes W\to V\otimes W is continuous.

Finally, a linear map of G-modules f:V\to W is said to be Gequivariant if

g\in G, v\in V \implies f(gv) = g\cdot f(v).

Note that this is equivalent to saying G acts trivially on f\in \text{Hom}_{\mathbb C}(V, W).


Compact Groups

From now till the end of the next article, G is assumed to be compact and Hausdorff; this case will be instrumental to the general theory even though GL_n\mathbb{C} is not compact. Here are some common examples of compact topological groups in representation theory:

\begin{aligned} U_n &= \{ M \in GL_n \mathbb{C} : M \overline{M}^t = I\},\\ SU_n &= \{ M \in GL_n\mathbb{C} : M\overline{M}^t = I, \det(M) = 1\},\\ O_n &=\{ M\in GL_n\mathbb{R} : M M^t = I\}, \\ SO_n &= \{M \in GL_n\mathbb{R} : MM^t = I, \det(M) = 1\}, \\ T &= \{ z \in \mathbb{C} : |z| = 1\}.\end{aligned}

These groups are compact because they are closed and bounded subsets of \mathbb{C}^{n\times n}. E.g. the condition for M \in U_n can be written out as a sequence of quadratic equations in \text{Re}(z_{ij}) and \text{Im}(z_{ij}) for 1\le i,j\le n. Thus the set U_n is closed; it is bounded since the condition gives:

|z_{i1}|^2 + |z_{i2}|^2 + \ldots + |z_{in}|^2 = 1, \quad\text{for } 1\le i\le n

so each |z_{ij}| \le 1.

Haar Measure

To proceed, we will borrow a result from harmonic analysis:

Theorem. If G is a compact Hausdorff group, then there is a measure m on G (called the Haar measure) satisfying:

  • m(G) = 1;
  • m is “left-invariant”, i.e. for any continuous f:G\to \mathbb{C}, we have:

\displaystyle  y\in G\implies \int_{x\in G} f(yx)\cdot dm(x) = \int_{x\in G} f(x)\cdot dm(x),

The Haar measure is unique and in fact right-invariant, i.e. \int_{x\in G} f(xy)\cdot dm(x) = \int_{x\in G} f(x) \cdot dm(x). We will implicitly assume the Haar measure is used when integrating over G.


The Haar measure is usually defined for locally compact Hausdorff groups in general, in which case we have to modify the statements a little. In particular, left-invariant measures are not necessarily right-invariant.


Semisimplicity of Continuous Representations

The following result is crucial.

Theorem. Let V be a G-module. If W\subseteq V is a G-invariant subspace, then we can find a G-invariant subspace W' \subseteq V such that V = W\oplus W'.


Let \pi : V\to V be any projection map onto W, so \pi^2 = \pi and \text{im}(\pi) = W. Now define:

\pi' : V\to V, \quad \pi'(v) := \int_{x\in G} x\pi(x^{-1}v)\cdot dx.

[Explicitly, one may take an isomorphism V\cong \mathbb{C}^N so each v\in V is a tuple of complex numbers; the above integral can be carried out component-wise, as N integrals of continuous functions.]

We then have:

  • The image of \pi' is in W, since \pi(V)=W and x(W)\subseteq W for x\in G.
  • If v\in W then x^{-1}v \in W\implies x\pi(x^{-1}v) = xx^{-1}v = v and \pi'(v) = v.
  • \pi'^2 = \pi': indeed if v\in V then \pi'(v) \in W by the first property and so \pi'(\pi'(v)) = \pi'(v) by the second.
  • The map \pi' is G-equivariant: for y\in G we have \pi'(yv) = y\pi'(v). Indeed by left-invariance we have: \pi'(v) = \int_{x\in G} yx \pi(x^{-1}y^{-1}v) dx. Replacing v with yv gives us the desired outcome.

Hence, \pi' is projection onto W and its kernel W’ satisfies V = W \oplus W'. Since \pi' is G-equivariant, W’ is a G-invariant subspace. ♦

Thus, every continuous representation of G can be written as a direct sum of irreps. Next, we have:

Schur’s Lemma. If V, W are irreps of G, then:

\text{Hom}_{\mathbb{C}[G]}(V, W) \cong \begin{cases} \mathbb{C}, \qquad &\text{if }V \cong W,\\ 0, \qquad &\text{otherwise.} \end{cases}

The proof is identical to that of the usual Schur’s lemma.


Suppose V\not\cong W and f:V\to W is G-equivariant. Then \text{ker}(f) \subseteq V and \text{im}(f)\subseteq W are subrepresentations, so \text{ker}(f) = 0, V and \text{im}(f) = 0, W. By considering various cases we have either

  1. \text{ker}(f) = 0, \text{im}(f) = W, in which case f is an isomorphism or
  2. \text{ker}(f) = V, \text{im}(f) = 0, in which case f = 0.

It remains to show that \text{Hom}_{\mathbb{C}[G]}(V,W) \cong \mathbb{C} when V\cong W.

It suffices to show that any isomorphism f:V\to V is a scalar: let v\in V be an eigenvector of with eigenvalue \lambda\in\mathbb{C} so f(v) = \lambda v. Then f - \lambda\cdot I : V\to V is G-equivariant and its kernel is non-zero; hence f - \lambda\cdot I = 0 \implies f = \lambda\cdot I. ♦


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Polynomials and Representations XXVII

From the previous article, we have columns jj’  in the column tabloid U, and given a set A (resp. B) of boxes in column j (resp. j’), we get:

  U-\beta_B(U) = U- \sum_{A,\, |A| = |B|} \alpha_{A,B}(U) \in \text{ker}(\pi)

where \alpha_{A,B}(U) is the column tabloid obtained by swapping contents of A with B while preserving the order. For example, we have the following modulo ker(π):


Here is our main result.

Main Theorem. Each of the following spans \ker\pi as a vector space:

  • the collection of U - \beta_B(U) over all sets B in column j’ and (j, j’) such that j < j’;
  • as above, except B comprises of the top k squares (for all k) of column j’ and j’ = j+1.

We have shown that the first set spans a subspace of ker(π); since the second set is a subset of the first, it remains to show that Q = ker(π), where Q is the vector space spanned by the second set.


Lemma. Consider [T] as T varies through all SYT with shape \lambda. These span the vector space \mathbb{C}[G]b_{T_0} / Q.


Create a new total ordering on the set of all fillings T of shape λ. Given T’T, we consider the rightmost column in which T’ and T differ; in that column, take the lowest square in which T’ and T differ; we write T’T if the entry in that square of T’ is larger than that of T:


Suppose T is a filling which is not an SYT. We claim that modulo Q, we can express [T] as a linear sum of [T’] for T’T.

  • Let T’ be obtained from T by sorting each column in ascending order; then [T’] = ±[T] and T’ ≥ T, so we assume each column of T is increasing.
  • If T is not an SYT, then we can find row i and column j of T such that T_{i, j} > T_{i,j+1}. Taking j’j+1 and the top i squares B of column j’ of T, we can replace [T] (modulo Q) by \beta_B([T]) = \sum_{|A|=|B|} \alpha_{A,B}([T]) where each term is [T’] for some T’>T.

Applying this iteratively, since there are finitely many fillings we eventually obtain any [T] as a linear combination of [T’] where T’ is SYT. ♦


Proof of Main Theorem

We know that Q\subseteq \text{ker}(\pi) so we get a surjection

\overline\pi : \mathbb{C}[G]b_{T_0}/Q \to \mathbb{C}[G]b_{T_0}a_{T_0} \cong V_\lambda.

Hence the dimension of \mathbb{C}[G]b_{T_0}/Q is at least \dim V_\lambda = f^\lambda. On the other hand, by our lemma above, the space is spanned by [T] for all SYT T so its dimension is at most the number of SYT of shape λ, i.e. f^\lambda. Thus \overline\pi is a bijection so Q = \text{ker}(\pi) as desired. ♦

As a side consequence, we have also proved that the set of [T] for all SYT T gives a basis for \mathbb{C}[G]b_{T_0}a_{T_0}.

This concludes our discussion of representations of the symmetric group S_d. Our next target is the representation theory of the general linear group GL_n \mathbb{C}.

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Polynomials and Representations XXVI

Let us fix a filling T_0 of shape \lambda and consider the surjective homomorphism of \mathbb{C}[G]-modules

\pi : \mathbb{C}[G]b_{T_0} \to \mathbb{C}[G]b_{T_0} a_{T_0} \subseteq \mathbb{C}[G]a_{T_0},

given by right-multiplying by a_{T_0}. Specifically, we will describe its kernel, which will have interesting consequences when we examine representations of GL_n\mathbb{C} later.

Row and Column Tabloids

By the lemma here, \mathbb{C}[G]a_{T_0} and \mathbb{C}[G]b_{T_0} can be described as follows.

For the former, take a basis comprising of “row tabloids“, i.e. for each filling T, take each row as a set, thus giving a partitioning [d] = A_1 \coprod A_2 \coprod \ldots. E.g. in the example below, the first two row tabloids are identical since they give rise to the same partitioning.


For \mathbb{C}[G]b_{T_0}, take a basis of “column tabloids“, by taking a filling T and taking its columns as sets, with the condition that swapping two elements in the same column flips the sign. E.g.



  • Given a filling T, the corresponding row tabloid is denoted by {T} and the row tabloid is [T].
  • If g\in S_d we have g(\{T\}) = \{g(T)\} and g([T]) = [g(T)]. E.g. (1,2) takes the above column tabloid to its negative while (1,2)(3,4) leaves it invariant.
  • Let us describe explicitly the correspondence between a column tabloid and an element of \mathbb{C}[G]b_{T_0}. For any column tabloid U = [T], where T is a filling, write T = g(T_0) for a unique g\in S_d. The corresponding element is gb_{T_0} \in \mathbb{C}[G]b_{T_0}.
    • The reader may worry that we left out the twist \chi(g), but you may check that this is consistent when we swap two elements in the same column.
  • For a row tabloid, we take \{g(T_0)\} to the element ga_{T_0} \in \mathbb{C}[G]a_{T_0}.


Column Swaps

Let us define some operations on column tabloids. Consider a column tabloid U := [T].

Suppose, in U, we have a (possibly empty) set B of boxes in column j’ and a set A of squares in column j such that |A| = |B|, with j’j. Let \alpha_{A,B}(U) denote the resulting column tabloid obtained by swapping squares A and B in order.


Now given a set B of squares in column j’ of U, define a linear map

\displaystyle \beta_B : \mathbb{C}[G]b_{T_0} \to \mathbb{C}[G]b_{T_0},\qquad \beta_B(U) := \sum_A \alpha_{A,B}(U),

where A runs through all sets of squares in column j of U such that |A| = |B|. Next, given a set Y of squares in column j’ of U, let:

\displaystyle\gamma_Y(U) := \sum_{B\subseteq Y} (-1)^{|B|}\beta_B(U) = \sum_{\substack{A,B\\ B\subseteq Y,\, |A| = |B|}}(-1)^{|B|}\alpha_{A,B}(U).

Clearly \gamma_Y(U) is an integer linear combination of the \beta_B(U); the following shows that the converse is true.


Lemma 1. For each set Y' of squares in column j', we have:

\displaystyle\sum_{Y \subseteq Y'} (-1)^{|Y|}\gamma_{Y}(U) = \beta_{Y'}(U).


Fix A, B satisfying B\subseteq Y' and |A| = |B|; we keep track of the coefficient of \alpha_{A,B}(U) in the expansion of LHS. The term is included in \gamma_Y(U) for all Y satisfying B\subseteq Y \subseteq Y'. Its coefficient is thus:

\displaystyle(-1)^{|B|}\cdot \sum_{Y,\, B\subseteq Y\subseteq Y'} (-1)^{|Y|}.

If B = Y’, this is 1; otherwise, fixing some y\in Y' - B, there is a 1-1 correspondence between Y containing y and Y not containing y; these cancel each other out so the sum is 0. Hence the overall sum is:

\displaystyle \sum_{A,\, |A| = |Y'|} \alpha_{A, Y'}(U) = \beta_{Y'}(U)

as desired. ♦


Lemma 2. We have \gamma_Y(U) \in \text{ker}\pi if Y is a non-empty set of squares in the j’-th column of U.


Let K\le H\le S_d be subgroups defined as follows:

  • H is the set of g \in S_d acting as the identity outside Y or the j-th column of U;
  • K is the set of g\in H taking Y\to Y and the j-th column of U back to itself.

For each g \in K we have \chi(g) g(U) = U so \sum_{g\in K} \chi(g) g(U) = |K|U.

Let h\in S_d run through all swaps of a subset A of the j-th column with B\subseteq Y preserving the order; this gives a complete set of coset representatives for H/K. On the other hand, the sum of these h(U) gives us \gamma_Y(U) so:

\displaystyle \sum_{g'\in H} \chi(g')g'(U) = \text{positive multiple of } \gamma_Y(U).

It remains to show that the LHS is in ker(π).

Fix filling T such that U = [T]. Unwinding the definition for map π, we first take g\in S_d such that T = g(T_0) then take the row tabloid for gb_{T_0}. Now, gb_{T_0} = gb_{T_0} g^{-1} g = b_T g so

\pi:[T] \ \mapsto\ b_{T}\{T\} = \sum_{g''\in C(T)} \chi(g'') g''\{T\}.

Since \pi is \mathbb{C}[G]-linear, we now need to show:

\displaystyle \sum_{g'\in H} \chi(g') g' \sum_{g''\in C(T)} \chi(g'')g''\{T\} \overset{?}=0.\quad (*)

Since Y\ne\emptyset, pick a square in it and a square to its left in the j-th column (since jj’). Fix the element g''\in C(T) and let h\in H be the transposition that swaps those two squares in g''T so h(g''\{T\}) = \{g''T\}. Taking g', g'h only in the outer sum of (*) gives:

(\chi(g')g' + \chi(g'h)g'h) \chi(g'') \{g''T\} = \chi(g')\chi(g'')g'(e - h) \{g''T\} = 0.

Summing over all cosets of \{e,h\} \le H and g'' \in C(T) we see that (*) sums to zero. ♦

From lemmas 1 and 2, we have:

Corollary. We have:

\beta_{Y'}(U) - \gamma_\emptyset(U) = \beta_{Y'}(U) - U\in \text{ker}(\pi).

E.g. modulo ker(π), we have:


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