# Questions tagged [characters]

For questions about the algebraic concept of 'character': a function from a group into a field satisfying certain properties. Not to be confused with the more commonly known psychological term.

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### Which finite simple groups are rational-relative-real?

A finite group $G$ is called rational if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$. Analogously, I'll call $G$ real ...
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### Interplay between additive and multiplicative characters of fields

Let $F$ be a countable field. Given a double Folner sequence $(F_N)_{N\in \mathbb{N}}$ in $F$, an additive character $\chi$ (i.e. $\chi: F \to \mathbb{S}_1$ satisfies $\chi(u+v)=\chi(u)\chi(v)$, for ...
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### Existence of a specific family of functions on an abelian group with vanishing properties on rank 2 subgroups

Fix a prime $p$, and let $W_0\subset W$ be an inclusion of a codimension one $\mathbb{F}_p$ vector spaces. Let $W_e$ denote a fixed nontrivial coset of $W_0$ in $W$. The question is whether there ...
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### Large values of characters of the symmetric group

For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the ...
1 vote
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### Demazure character (in type A) from Kostant's partition function?

The Kostka coefficients are the coefficients of Schur expanded in monomial basis, i.e., $s_\lambda = \sum_\mu K_{\lambda,\mu} m_\mu$. They are also the coefficients obtained by taking the complete ...
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### Sum of weights of an irreducible representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
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Let $G$ be a finite abelian group, $X$ and $Y$ be two non-empty subsets of $G$ of equal size. Suppose that for each irreducible character $\chi$ of $G$ we have $\sum_{x\in X}\chi(x)=\sum_{y\in Y}\chi(... 4 votes 0 answers 221 views ### Orbits of group representation over$\mathbb{F}_2$Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ... 12 votes 2 answers 1k views ### The character table of the symmetric group modulo m Let$S_n$be the symmetric group and$M_n$the character table of$S_n$as a matrix (in some order) for$n \geq 2$. Question: Is it true that the rank of$M_n$as a matrix modulo$m$for$m \geq 2$... 3 votes 0 answers 167 views ### Characterizing polynomials which behave like a logarithm modulo$1$This is a version of a question I asked in Math StackExchange about two weeks ago, which is still unanswered. (UPDATE: the original question for$R=\mathbb{Z}$has been finally answered, but its ... 1 vote 0 answers 124 views ### A question about Theorem 2.3.1 in Tate's thesis [closed] I don't understand how to prove a conclusion in the Theorem. When k is$p$-adic, the subgroups 1+$p^{v}$,$v>0$, of$u(|u|=1)$form a fundamental system of neighborhoods of$1$in$u$, We must ... 1 vote 0 answers 154 views ### Restriction of a line bundle on$G/B$to a fibre which is isomorphic to$\mathbb{P}^1$Let$G$be a reductive group over a field$k$of characteristic zero with maximal split torus$T$, Borel$B \supset T$and Weyl group$W$. Set$X:=G/B$and$C_w:=BwB/B \subset X$for$w \in W$the ... 3 votes 2 answers 299 views ### Character which defines canonical bundle on flag variety Let$G$be a reductive group over a field$k$of characteristic zero with maximal split torus$T$and Borel$B \supset T$defining a set of simple roots$\Delta$. Additionally let$\rho$be the half ... 0 votes 0 answers 50 views ### Counting the number of generating triples of various types in finite simple groups I am trying to figure out how specific generating triples in finite simple groups are calculated. My understanding is that it uses Frobenius's formula and character theory. I'm not an expert on ... 17 votes 2 answers 764 views ### The finite groups with a zero entry in each column of its character table (except the first one)$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups$G$having a zero entry in each column of its character table (except the first one), i.e. for all$g \...
Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well ...