25
votes
On permanents and determinants of finite groups
I won't discuss the permanent part of the question, but I think the other part can be done easily, even without Galois theory. Let $X = X(G)$ denote the character table of $G$ (rows indexed by complex ...
- 44.4k
23
votes
Explicit character tables of non-existent finite simple groups
This is not really a proper answer, but it's way too long for a comment:
My understanding is that by the time a complete character table has been obtained, this is very strong evidence for the ...
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21
votes
The character table of the symmetric group modulo m
This is true when $m$ is prime and false in general.
Counterexample. Take $S_8$ with $m=6$. Computer calculations show that the $\mathbb{Z}$-rank of the character table of $S_8$ with entries taken ...
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21
votes
The character table of the symmetric group modulo m
When $m$ is prime there is a simpler proof. The Smith normal form of the character table of $S_n$ is computed at Problem 14 here (solution here). From this it follows that the rank of the character ...
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19
votes
Accepted
Explicit version of the Burgess theorem
There are now at least two instances of such an explicit result.
Theorem 1.1 of Bordignon: https://arxiv.org/abs/2001.05114.
Theorem 1.1 (and Corollary 1.2) of Jain-Sharma, Khale, and Liu: https://...
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17
votes
Accepted
What did Frobenius prove about $M_{12}$?
It seems to me that Frobenius is using lots of specific facts about the permutation group $M_{12}$ (and $M_{24}$, respectively) here, and that he has no doubts about the existence of these groups. In ...
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16
votes
What did Frobenius prove about $M_{12}$?
This summary of Frobenius' 1904 paper might be of use:
Thomas Hawkins, The Mathematics of Frobenius in Context (page 527-528).
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16
votes
Accepted
Tannakian Formalism for the Quaternions and Dihedral Group
Let $V_D$ and $V_Q$ be the two dimensional simple representations of $D_4$ and $Q_8$ respectively. Let $1_D$ and $1_Q$ denote their trivial representations.
Suppose that there is a tensor equivalence ...
- 8,866
16
votes
Accepted
The finite groups with a zero entry in each column of its character table (except the first one)
Partial answer: the finite group $G$ is clearly in this class if it has a $p$-block of defect zero for every prime $p$ which divides $|G|$. This is a sufficient condition which may not be necessary. ...
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16
votes
Accepted
Finite groups with integral character table
There is no complete classification, but some structural results are known. To give you something to search for: such groups are called $\mathbb{Q}$-groups. There is a whole book devoted to their ...
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15
votes
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Conductor as volume of the integers ring
Apply the Fourier Inversion Formula to the characteristic function $\Phi(x) = \chi_\mathcal{O}(x)$ of the ring $\mathcal{O}$ of integers in $F$. The Fourier transform is the integral $\widehat{\Phi }(...
- 11.2k
14
votes
A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (III)
Here is a proof of (i):
Since the relevant primes $p$ are $\equiv 1 \bmod 4$, we have
$S_p(c,d) = \frac{1}{2} T_p(c,d)$, where
$$ T_p(c,d) = \sum_{x=0}^{p-1} \left(\frac{x^5+cx^3+dx}{p}\right) \,. $$
...
- 11.3k
13
votes
The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group
Here are a few remarks which may be helpful. I may be able to say more later. The condition that $\operatorname{cp}(G) > \frac{1}{4}$ already severely restricts the possibilities for $G$. If $G$ ...
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12
votes
Tannakian Formalism for the Quaternions and Dihedral Group
The categories ${\rm Rep}(Q_8)$ and ${\rm Rep}(D_8)$ are not equivalent as tensor categories. They have the same Grothendieck ring, but they have non equivalent associators. As far as I am aware, it ...
- 3,830
12
votes
Accepted
Is a $G$-invariant character $\theta$ of $H$ extendible to $G$?
The answer is yes (and some of the comments were moving in the right direction): Let $T$ be a transversal to $H$ in $G,$ and let $\sigma$ afford the representation of $H.$ For each $t \in T,$ there is ...
- 44.4k
12
votes
Closed formulas for the character of the symmetric group
Giving explicit formulas for the characters is the content of the recent article "An explicit formula for the characters of the symmetric group" by
Michel Lassalle :
https://link.springer.com/...
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12
votes
Specific application of Cauchy-Schwarz and Large Sieve
The answer of Ofir Gorodetsky is perfectly fine, but one can also apply the Cauchy-Schwarz inequality for $L^2$ spaces directly.
Indeed, let us consider the $L^2$ space of functions on the set of ...
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11
votes
Accepted
About the existence of characters on $B(X)$
I guess that you mean that $B(H)$ has no character (=continuous unital algebra homomorphism into $\mathbf{C}$) if $H$ has dimension $\neq 1$ (idem for $M_n(\mathbf{C})$ for $n\neq 1$), and thus that ...
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11
votes
Accepted
On the structure of a finite group of order $144$
Since $G$ has an irreducible character of degree $9 = |G|_{3},$ we have $O_{3}(G) = 1,$ so $G$ has more than one Sylow $3$-subgroup. If $G$ has only $4$ Sylow $3$-subgroups, then $G$ has a normal ...
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10
votes
About the existence of characters on $B(X)$
Examples were known before the Argyros-Haydon space mentioned in Yves Cornulier's answer. For instance, if $J$ denotes the James space, then the image of the canonical map $J\to J^{**}$ has ...
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10
votes
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Are the character degrees determined by the conjugacy class sizes?
SmallGroup(128,227) and SmallGroup(128,731)) are counterexamples.
...
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10
votes
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Pólya–Vinogradov inequality for Eisenstein integers
No.
Such a bound would imply a similar bound on
$$\displaystyle \left \lvert \sum_{N(z) = M} \left(\frac{z}{w} \right)_3 \right \rvert.$$
If $M$ is a product of distinct primes $p_1,\dots p_n$ ...
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9
votes
On permanents and determinants of finite groups
I'll address the remaining question 2.
Theorem: If $|G|=4r+2$ then the permanent of its character table vanishes.
Proof: We know $G$ has a normal subgroup $H$ of index $2$. Let's denote the cosets of $...
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8
votes
Accepted
Is there a converse to the Brauer–Nesbitt theorem?
Not always — e.g. $g(1)$ should be an integer. The desired description is given in
Helling, H., Eine Kennzeichnung von Charakteren auf Gruppen und assoziativen Algebren, Commun. Algebra 1, 491-501 (...
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8
votes
What did Frobenius prove about $M_{12}$?
Not really an answer, and probably contained in Frobenius's paper, but a starting point might be that (using a result now usually credited to Blichfeldt) if $G$ is a sharply $5$-transitive group (of ...
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8
votes
$G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?
The answer is Yes by brute forcing. As there are only 208 groups of order 240, we can check them one by one in GAP:
...
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8
votes
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$G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?
$G$ is non-solvable, so must have $A_{5}$ as a composition factor (as no other non-Abelian simple group has less than $168$). Hence $F(G)$ can have order at most $4$.
If $G$ has no component, then $F(...
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8
votes
Accepted
Why is Nagao's theorem the "Module theoretic version of Brauer's second main theorem"?
To see the connection, it is easiest to work over a local ring $R$ of characteristic zero with residue field $R/J(R) \cong \mathbb{F}$ (there are some technicalities I am omitting here for the sake ...
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8
votes
Character theory and Quantum Chemistry
Introducing groups into quantum theory, by Erhard Scholz (2006):
In quantum chemistry, representations of permutation groups made their
first appearance about the same time as they did in ...
- 188k
8
votes
Accepted
Generalization of $\lim_{n \rightarrow \infty} \prod_{i=1}^{n}\frac{2i-1}{2i}$ for a character $\chi:\mathbb{Z}/s \mathbb{Z} \rightarrow \mathbb{C}^*$
Summary: I consider the limit $\lim_{n\to\infty}\prod_{i=1}^n i^{\chi(i)}$ (let me drop the $\mod s\mathbb Z$ for brevity). If we restrict to $n$ divisible by $s$, then the limit will always be equal ...
- 28.2k
Only top scored, non community-wiki answers of a minimum length are eligible