Monthly Archives: December 2015

Splitting fields of representations of generalized symmetric groups, 4

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First a technical definition.

Let A=C_\ell^n. Let \eta_k(z)=z^k, for z\in C_\ell and 1\leq k\leq \ell-1. For \eta\in C_\ell^*, let \mu\otimes \eta =(\mu_1\eta,\mu_2\eta,...,\mu_n\eta) where \mu=(\mu_1,\mu_2,...,\mu_n). This defines an action of C_\ell^* on A^* and hence on the set of equivalence classes of G, G^*. We call two representations \theta_{\mu,\rho}, \theta_{\mu',\rho'} C_\ell^*-equivalent, and write

\theta_{\mu,\rho}\sim_\ell \theta_{\mu',\rho'},

if \rho=\rho' and \mu'=\mu\otimes \eta for some \eta\in C_\ell^*. Similarly, we call two characters \mu, \mu' of C_\ell^n C_\ell^*-equivalent, and write

\mu\sim_\ell \mu',

if \mu'=\mu\otimes \eta for some \eta\in C_\ell^*.

For example, Let \ell =9, n=3 and \mu=(\eta_2,\eta_5,\eta_8). Then \mu\sim \mu\otimes\eta_3.

Let \theta_{\mu,\rho} be as in the previous post. Note that

\theta_{\mu\otimes \eta,\rho} = \theta_{\mu,\rho}\otimes \eta ,

for \eta\in C_\ell^*. Therefore, the matrix representations of two C_\ell^*-equivalent representations differ only
by a character.

Let G=C_\ell^n\, >\!\!\lhd \, S_n.

The results in the above section tells us how to construct all the irreducible representations of G. We must

  1. write down all the characters (i.e., 1-dimensional representations) of A=C_\ell^n,
  2. describe the action of S_n on A^*,
  3. for each \mu\in [A^*], compute the stabilizer (S_n)_{\mu},
  4. describe all irreducible representations of each (S_n)_{\mu},
  5. write down the formula for the character of \theta_{\mu,\rho}.

Write \mu\in [A^*] as \mu=(\mu_1,...,\mu_n), where each component is a character of the cyclic group C_\ell, \mu_j\in C_\ell^*. Let \mu'_1,...,\mu'_r denote all the distinct characters which occur in \mu, so

\{\mu'_1,...,\mu'_r\}=\{ \mu_1,...,\mu_n\}.

Let n_1 denote the number of \mu'_1‘s in \mu, n_2 denote the number of \mu'_2‘s in \mu, …, n_r denote the number of \mu'_r‘s in \mu. Then n=n_1+...+n_r. Call this the partition associated to \mu.

If two characters \mu=(\mu_1,...,\mu_n), \mu'=(\mu'_1,...,\mu'_n) belong to the same class in [(C_\ell^n)^*], under the S_n-equivalence relation, then their associated partitions are equal.

The Frobenius formula for the character of an induced representation gives the following character formula. Let \chi denote the character of \theta_{\mu,\rho}. Then

\chi(\vec{v},p)=\sum_{g\in S_n/(S_n)_\mu} \chi^o_\rho(gpg^{-1})\mu^g(\vec{v}),

for all \vec{v}\in C_\ell^n and p\in S_n. In particular, if p=1 then

\chi(\vec{v},1)=({\rm dim}\ \rho)\sum_{g\in S_n/(S_n)_\mu} \mu^g(\vec{v}).

Splitting fields of representations of generalized symmetric groups, 3

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The representations of a semi-direct product of a group H by an abelian group A, written G=A\, >\!\!\lhd \, H (so A is normal in G) can be described explicitly in terms of the representations of A and H. The purpose of this post is to explain how this is done.

Again, a good reference for all this is Serre’s well-known book, Linear representations of finite groups.

Let f be a class function on $H$. Extend f to G trivially as follows:

f^0(g)= \left\{ \begin{array}{cc} f(g),&g\in H,\\ 0, & g\notin H, \end{array} \right.

for all g\in G. This is not a class function on G in general. To remedy this, we “average over G” using conjugation: Define the function f^G=Ind_H^G(f) induced by f to be

Ind_H^G(f)(g)={1\over |H|}\sum_{x\in G} f^0(x^{-1}gx)=\sum_{x\in G/H}f^0(x^{-1}gx).

This is referred to as the Frobenius formula.

Since A is normal in G, G acts on the vector space of formal complex linear combinations of elements of A^* (=the characters of A),

V={\mathbb{C}}[A^*]=span\{\mu\ |\ \mu\in A^*\},

by

(g\mu)(a)=\mu(g^{-1}ag),\ \ \ \ \forall g\in G,\ a\in A,\ \mu\in A^*.

We may restrict this action to H, giving us a homomorphism \phi^*:H\rightarrow S_{A^*}, where S_{A^*} denotes the symmetric group of all permutations of the set A^*. This restricted action is an equivalence relation on A^* which we refer to below as the H-equivalence relation}. Let [A^*] denote the set of equivalence classes of this equivalence relation. If \mu,\mu' belong to the same equivalence class then we write

\mu'\sim \mu

(or \mu'\sim_H\mu if there is any possible ambiguity). When there is no harm, we identify each element of [A^*] with a character of A.

Suppose that H acts on A by means of the automorphism given by a homomorphism \phi:H\rightarrow S_{A}, where S_{A} denotes the symmetric group of all permutations of the set A. In this case, two characters \tau,\tau'\in A^* are equivalent if there is an element h\in H such that, for all a\in A, we have \tau'(a)=\tau(\phi(h)(a)).

For each \mu\in [A^*], let

H_{\mu}=\{h\in H\ |\ h\mu = \mu\}.

This group is called the stabilizer of \mu in H. Let

G_{\mu}=A\, >\!\!\lhd \, H_{\mu},

for each \mu\in [A^*]. There is a natural projection map

p_{\mu}:G_{\mu}\rightarrow H_{\mu}

given by ah\longmapsto h, i.e., by p_\mu(ah)=a.

Extend each character \mu\in [A^*] from H_{\mu} to G_{\mu} trivially by defining

\mu(ah)=\mu(a),

for all a\in A and h\in H_{\mu}. This defines a character \mu\in G^*_{\mu}. For each \rho\in H_{\mu}^*, say \rho:H_{\mu} \rightarrow Aut(V), let \tilde{\rho}\in G_{\mu}^* denote the representation of G_{\mu} obtained by pulling back \rho via the projection p_\mu:G_{\mu}\rightarrow H_{\mu}, i.e., define

\tilde{\rho}=\rho\circ p_{\mu}.

For each \mu \in [A^*] and \rho\in H_\mu^* as above, let

\theta_{\mu,\rho}=Ind_{G_\mu}^G(\mu\cdot \tilde{\rho}).

Finally, we can completely describe all the irreducible representations of G=A\, >\!\!\lhd \, H. (This is Proposition 25 in chapter 8 of Serre’s book.)

Theorem:

  1. For each \mu \in [A^*] and \rho\in H_\mu^*, as above, then \theta_{\mu,\rho} is an irreducible representation of G.
  2. Suppose \mu_1,\mu_2 \in [A^*], \rho_1\in H_{\mu_1}^*, \rho_2\in H_{\mu_2}^*. If \theta_{\mu_1,\rho_1}\cong \theta_{\mu_2,\rho_2} then \mu_1\sim \mu_2 and \rho_1\cong \rho_2.
  3. If \pi is an irreducible representation of G then \pi\cong \theta_{\mu,\rho}, for some \mu \in [A^*] and \rho\in H_{\mu}^* as above.

In the next post, we will examine the special case A=C_\ell^n and H=S_n.

Splitting fields of representations of generalized symmetric groups, 2

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In general, there are three types of (complex) representations of a finite group G. (A good reference for all this is Serre’s well-known book, Linear representations of finite groups.)

Let \rho:H\rightarrow Aut(W) be an n-dimensional irreducible representation of a finite group G on a complex vector space W. Let \chi denote the character of \rho.

Exactly one of the following possibilities must hold:

  • One of the values of the character \chi is not real. Such representations will be called complex (or type 1 or unitary).
  • All the values of \chi are real and \rho is realizable by a representation over a real vector space. Such representations will be called real (or type 2 or orthogonal).
  • All the values of \chi are real but \rho is not realizable by a representation over a real vector space. Such representations will be called quaternionic (or type 3 or symplectic).

Proposition (Frobenius-Schur): Let \rho:H\rightarrow Aut(W) be an irreducible representation of a finite group G on a complex vector space W with character $\chi$. Then

{1\over |G|} \sum_{g\in G}\chi(g^2)= \left\{ \begin{array}{cc} 0,&\rho\ {\rm complex},\\ 1,&\rho\ {\rm real},\\ -1,&\rho\ {\rm quaternionic}. \end{array} \right.

This quantity is sometimes called the Frobenius-Schur indicator of \rho.

It can be shown that if \rho \rho'\cong \rho are equivalent representations then \rho and \rho' have the same type.

In the next post we will examine the types that the irreducible representations of semi-direct product G=C_\ell^n\, >\!\!\lhd \, S_n fall into.

Splitting fields of representations of generalized symmetric groups, 1

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The main result of this series of blog posts (originally written 1998), which is expository, is to determine (in a sense made precise later) the splitting field of any irreducible character of a generalized symmetric group. This was basically solved by M. Benard in a 1976 J. of Algebra paper. We use one of his results to make the splitting field explicit.

Notation and definitions

Let C_\ell denote the cyclic group of order \ell\geq 1, let S_n denote the symmetric group of degree n\geq 1, and let G denote the semi-direct product G=C_\ell^n\, >\!\!\lhd \, S_n. We think of this as the set of pairs (v,p), with

  • v=(v_1,...,v_n), where each v_i\in C_\ell=\{0,1,...,\ell-1\},
  • p\in S_n,
  • S_n acts on C_\ell^n by p(v)=(v_{p(1)},v_{p(2)},...,v_{p(n)}),
  • multiplication given by (v,p)*(v',p')=(v+p(v'),pp'),\ \ \ \ \ (v,p),(v',p')\in G.

A group of the form C_\ell^n\, >\!\!\lhd \, S_n, also written S_n \ {\rm wr}\ C_\ell (here wr denotes the wreath product), will be called a generalized symmetric group.

We may identify each element of C_\ell^n\, >\!\!\lhd \, S_n with an n\times n monomial matrix (a matrix with exactly one non-zero entry in each row and column) with entries in C_\ell.

 

Our main goal will be to investigate the following question: What is the smallest extension of {\mathbb{Q}} required to realize (using matrices) a given irreducible character of a generalized symmetric group?