For questions about characters (homomorphisms from a group into the multiplicative group of a field).

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2
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1answer
57 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, \operatorname{sgn}...
1
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0answers
38 views

Is character of a group representation the same as trace?

If so, why cannot the Klein group's character be zero? The group element of Klein group matrices can be traceless, right?
2
votes
1answer
59 views

A reference for the equality $|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$

Let $G$ be a finite group and $H$ a subgroup. Let $V_1, \dots , V_r$ be (equivalence class representatives for) the irreducible complex representations of $G$. Let the stabilizer subspace $V_i^H = \{ ...
2
votes
1answer
82 views

Sum of sum of elements in conjugacy class is a multiple of them if and only if $G=G'$

I have another question on character/group theory. This one seems to be a bit harder. Let $Cl(g_1),...Cl(g_r)$ be the conjugacy classes of a finite group, $G$ and let $C_i \in \mathbb{C}(G)$ (the ...
2
votes
1answer
50 views

Faithful character of degree less than p gives abelian p-Sylow groups.

I am trying to prove: Suppose $p$ divides $|G|$ and let $\chi \in Irr(G)$ if $\chi$ is faithful and its degree is less than $p$ then any $p$-Sylow subgroup of $G$ is abelian. I have tried to ...
0
votes
1answer
72 views

Character of $Sym^2(V)$ and decomposition into irreducible representations

Let $G=S_3$ be the symmetric group on three elements, whose character table is given as follows: Let $V$ be the unique irreducible representation of dimension $2$ Question 1: Compute the character ...
2
votes
1answer
29 views

Character regular representation

Consider the regular representation of a finite group $G$ and let $X_{reg}$ be its character. Let $(\pi, V)$ be any finite dimensional representation of $G$ with character $X$. Show that $<X_{reg}...
0
votes
1answer
60 views

Character Table, Row and Column orthogonality, Conjugacy Classes

Let $G$ be a finite group with conjugacy classes $C_1, C_2, ..., C_k$ and let $g_i \in C_i$ be an element for each $i=1, ..., k$ Part 1: State the theorems on row and column orthogonality in the ...
1
vote
1answer
39 views

Wedderburn component of $\mathbb C[G]$ corresponding to a contragredient character

Let $G$ be a finite group and let $\phi$ be an irreducible character of $G$ over $\mathbb C$. How does the Wedderburn component of $\mathbb C[G]$ corresponding to the contragredient character of $\phi$...
2
votes
1answer
35 views

Let ($\pi, V$) be a representation of $G$ with character $X$: if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations

Let $(\pi, V)$ be a representation of $G$ with character $X$. Prove that if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations I was under the impression that the inner ...
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0answers
21 views

Kloosterman Sums and K3 Surfaces

In R. Livne's Motivic Orthogonal Two-Dimensional Representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, he provides a technique for computing the $\zeta$-function of the $K3$ surface $$X=\{...
0
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0answers
33 views

Is this statement true? (characterize elements of dual group)

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
2
votes
1answer
14 views

Question on normal subgroup and sum of characters.

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Prove that $$\mid G:N \mid = \sum \chi(1)^2$$ where the sum ranges over all irreducible characters of $G$ such that $N \subseteq \ker ...
1
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0answers
30 views

Is there a non-trivial character on any locally compact Abelian group?

Let $G$ be a locally compact Abelian group. Is there a non-trivial character on $G$? indeed, i want the proof of the existence of a non-trivial character on a locally compact Abelian group.
0
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0answers
32 views

Subspace generated by irreducible characters

It is mentioned in the answer for this question: Perhaps we should also note that the subspace generated by irreducible characters is the same as that generated by their conjugates. Can you ...
0
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0answers
26 views

Proof of existence of a non-trivial character on a local field of positive characteristic

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
1
vote
0answers
24 views

Hints in proving the irreducibility of the permutation representation of $S_n$

Can anyone give me any hints of how to proceed: I have scoured this forum and exhausted all online notes. I am trying to prove that there is no room for an irreducible permutation representation, $\...
0
votes
0answers
45 views

discontinuous group homomorphism from $(\mathbb{R},+)$ to $S^1$,unit circle

It is well known that if $\gamma:(\mathbb{R},+)\rightarrow S^1$ is continuous homomorphism,then $\exists y\in\mathbb{R}$,such that $\gamma(x)=e^{ixy}$. Show that there is a discontinuous homomorphism ...
0
votes
1answer
50 views

Hypothesis: $\sum\limits_{n \in A} ccl_p(n) = p-1 $ [closed]

Let G be a the finite set of integers $1, \dots , p-1 $ One can see, if $|G| = p-1 $ with $p$ prime, then $\forall a \in G: \exists n \in \mathbb{N}$ with $ a^n = 1$ mod $p$ and $ 1 \le n < p $ I ...
3
votes
1answer
38 views

A question concerning an exercise from Tao Vu

This is the exercise 4.1.5 from Tao Vu Additive Combinatorics. $Z$ is a finite additive group with a fixed symmetric non-degenerate bilinear form $\cdot$ Define $e: \mathbb{R}/\mathbb{Z} \to \mathbb{...
1
vote
1answer
27 views

Existence of homomorphism from $GO(2n,\mathbb{C})$ to $\mathbb{C}^*$ that doesn't lose too much information

L.S., Denote $GO(2n,\mathbb{C})$ the group of matrices that leave an inner product invariant up to a scalar. We have for all $A \in GO(2n,\mathbb{C}), v,w$ vectors, that: $\langle A(v),A(w)\rangle = \...
0
votes
1answer
46 views

Can one always construct a square-root of a homomorphism to $\mathbb{C}^*$?

L.S., Let $G$ a group and $\chi: G \rightarrow \mathbb{C}^*$ a morphism of groups. If we set $\delta: G \rightarrow \mathbb{C}$ by $\delta(g) = \sqrt{\chi(g)}$, where we choose one branch of $\mathbb{...
0
votes
0answers
64 views

The character group of $G$ for an abelian group $G$.

Problem Statement: Prove that the one-dimensional characters of a group $G$ form a group under multiplication of functions, i.e. where the group operation is: $$(\chi\cdot\chi')(g)=\chi(g)\chi'(g)$$...
0
votes
0answers
32 views

“linear independence” of characters? Artin's theorem.

Let $G$ be a monoid and $K$ be a field. We define a character of $G$ in $K$ as a monoid homomorphism $f\colon G\to K^{\times}$, where $K^{\times}$ denotes the multiplicative group of $K$. One fact ...
0
votes
0answers
38 views

About character table and normal subgroups

I have read that: All normal subgroups of a group $G$ can be recognized from its character table. The kernel of a character $\chi$ is the set of elements $g$ in $G$ for which $\chi(g) = \chi(1)$; ...
0
votes
0answers
16 views

Computing character for the partition $(2, 2, 1, 1)$ using Murnaghan–Nakayama rule

I am trying to understand an example from Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. The group ...
0
votes
0answers
25 views

Understanding the Murnaghan–Nakayama rule

I am trying to understand the Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. Here is the screenshot....
2
votes
0answers
29 views

Twisted Kloosterman Sums

A twisted Kloosterman sum is a character sum of the form $$S(\chi, \psi, \eta)=\sum_{t\in (\mathbb{F}_q)^{\times}} \chi(t) \psi(t) \eta(t^{-1}).$$ Here, $\chi$ is a multiplitive character of $(\mathbb{...
1
vote
2answers
22 views

Character Sums and Characteristic Functions

Let $\mathbb{F}_q$ be the finite field with $q$ elements with $q$ odd. Consider the subgroup $H$ of $(\mathbb{F}_q)^{\times}$ consisting of squares: $H=\{x^2\,:\,x\in(\mathbb{F}_q)^{\times}\}$. We can ...
1
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0answers
35 views

A character on a subgroup could be written as a difference with some character on the whole group, help on argumentation

Let $G$ be a finite group, and $U \le G$ of odd order such that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t$ and assume that $U^g \ne U$ implies $U^g \cap U = 1$. Then I can not ...
1
vote
1answer
41 views

On the decomposition of the group ring $\mathbb Q[G]$ over the rationals if $G$ is finite and cyclic

Let $G$ be a cyclic finite group of order $n$. I tried to determine the structure of the group ring $\mathbb Q[G]$ over the rationals $\mathbb Q$, what I got for even $n$ is $$ \mathbb Q[G] = A \...
3
votes
0answers
38 views

Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove this?...
0
votes
0answers
23 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
1
vote
4answers
48 views

Class function and character of $S_3$ representation

I particularly need help with question 2. The $\textit{character table}$ for $S_3$ is given as follows:$$\begin{array}{c|c|c|} & \text{1} & \text{(12)} & \text{(123)} \\ \hline \text{$...
3
votes
1answer
64 views

(Infinite) Non-abelian group with only linear characters

If $G$ is an abelian group, then every irreducible character has dimension one (i.e. is linear), for finite group we also have a converse. Do we have a converse for infinite groups? Or: Does there ...
3
votes
1answer
44 views

A direct proof that for finite $G$ we have “$G$ abelian iff all irreducible characters are linear”

A finite group is abelian iff all its irreducible characters have dimension one (hence are linear). A common proof uses that the number of irreducible representations equals the number of conjugacy ...
2
votes
1answer
31 views

Relating a Character sum to a Gauss sum

Let $q$ be a prime power. Consider the mapping $f:(\mathbb{F}_q)^{\times} \to (\mathbb{F}_q)^{\times},$ where $x\mapsto x^2$. I was interested in sums of the form $$\sum_{t\in \mbox{Im}(f)} \psi_a(t)\...
0
votes
0answers
30 views

Constructing character table of subgroup from character table of whole group

If $\psi : G \to \mathbb C$ is a character and $U \le G$, then $\psi_{|U} : U \to \mathbb C$ is a character, but I guess this might not be irreducible with respect to $U$. But is it possible to ...
0
votes
0answers
42 views

How to construct a character table? (E.g Klein 4 group)

Could someone explain to me how you make a character table? Say I wanted to give the character table for the Klein $4$ group, $K$. $K$ is isomorphic to the product $\mathbb{Z}/2\mathbb{Z} \times \...
1
vote
1answer
50 views

Conjugacy classes, irreducible representations, character table of $D_{10}$ (order 20)

$D_{10}=\langle r,s \rangle$ is the dihedral group of order 20 . I have been struggling a bit with this question, particularly c, regarding values in the character table (a). Find the conjugacy ...
0
votes
1answer
33 views

Why do we have these values of the generalized character when evaluated with the scalar product?

Let $U \le G$ be a subgroup of odd order of the finite group $G$. Suppose $t \notin U$ is an involution with $u^t \in uU'$ for all $u \in U$, where $U'$ denotes the commutator subgroup of $U$. Set $T :...
0
votes
1answer
37 views

Extending a linear character of $U$ to $TU$, where $T$ is generated by an involution normalising $U$

Let $U \le G$ be a subgroup of the finite group $G$ of odd order. And suppose $t \notin U$ is an involution normalising $U$, i.e. $U^t = U$ and $t^2 = 1$. Assume $t$ centralizes $U / U'$, i.e. $u^tU' =...
2
votes
1answer
50 views

Does same character table imply isomorphic abelianizations?

We know two finite groups with the same character table might not be isomorphic (e.g. $D_4$ and $Q_8$), but the sizes of their abelianizations are equal (in fact equal to the number of linear ...
1
vote
0answers
23 views

Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
1
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0answers
35 views

All the characters on $S^1$

Would anyone please give me a hint to find all the characters on $S^1$? By characters I mean complex valued functions on $ S ^ 1 $ that their values have norm $1 $ and they satisfy: $f (x+y)=f (x) ...
0
votes
0answers
34 views

Inverse of character table

The character table of a group is always invertible, because the rows are orthogonal. Is there a general formula to compute the inverse of the character table?
0
votes
2answers
41 views

Degree one irreducible representations

In section 2.5 of his Linear representations of finite groups (I have the french copy), Serre gives an example of determination of the character table of a group $G$. The group $G$ is taken to be $S_3$...
1
vote
0answers
17 views

Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
1
vote
1answer
49 views

Show that character vanishes on specific element $g$ if for $H \le G$ we have $[\chi_H, 1_H] = 0$ and all elements of $Hg$ are conjugate in $G$

Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show ...
2
votes
0answers
38 views

If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...