All Questions
15 questions
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The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
1
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1
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206
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Reflections on subspaces of $\text{codim} > 1$
Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$.
Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
- 1,813
4
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1
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211
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Nonempty intersection of cosets of finite-index subgroups
$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE.
Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq ...
- 10.6k
1
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69
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Convolutions of (m)-associahedra and (m)-noncrossing partition polynomials--combinatorial proofs?
I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II).
(Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...
- 10.5k
4
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108
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Doubly stochastic matrices that remain doubly stochastic after conjugating by the character table of a finite abelian group
I am curious if anything is known about the following.
Let $\Gamma$ be a finite abelian group, and let $\chi$ be its character table, normalized so that it is a unitary matrix. E.g., if $\Gamma$ is $\...
- 2,339
7
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1
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326
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Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$
Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
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Simultaneous similarity classes of pairs in $\mathrm{GL}_{n}(\Bbb Z / p\Bbb Z)$?
$\DeclareMathOperator{\GL}{\operatorname{GL}}$Let $G$ be an elementary abelian $p$-group of rank $2$. Let $\alpha, \beta :G\rightarrow \GL_{n}(\Bbb Z / p\Bbb Z)$ be two injective homomorphisms. The ...
- 181
1
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519
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How typical are integer isometries on a hypercube? Littlewood-Offord problem for Bernoulli Gram matrices
Let $m\geq 3$ be fixed and $n\to\infty$. Consider $v=(v_j)_{j\leq m}$ with $v_1,\ldots,v_m\in \{-1,+1\}^n$. Let:
$N_I(v)$ be the number of sequences $u_1,\ldots,u_m\in \{-1,+1\}^n$ isometric to $v$ ...
- 175
2
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127
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When an isometry is a hypercube symmetry?
Suppose that sequences $v_1,\ldots,v_m\in \{-1,+1\}^n$ and $u_1,\ldots,u_m\in \{-1,+1\}^n$ are isometric in $\mathbb{R}^n$, i.e. $u_j=Qv_j$ for some orthogonal matrix $Q$. What is the largest $m$ such ...
- 175
1
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0
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127
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Lower and upper bounds of the distance between two Frobenius numbers
I consider two sequences of numbers: $A=\{a_1,...,a_{m-1},n\}$ and $B=\{n-a_{m-1},...,n-a_1,n\}$, where $a_1 < a_2 < ... < a_{m-1} < n$ and $\gcd(A) = \gcd(B) = 1$.
I investigate the ...
- 13
15
votes
1
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679
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Submodules of $({\mathbb Z}/6{\mathbb Z})^n$ intersecting $\{0,1\}^n$ trivially
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\Z}{{\mathbb Z}}$
Suppose that $\F$ is a finite field of prime order $p:=|\F|$, and let $n$ be a positive integer. I consider the regime where $\F$ is ...
- 23k
5
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1
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503
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does this set of permutations form a group? And more
Consider the group of $mn\times mn$ permutation matrices $\mathfrak{S}_{mn}$ and partition each such matrix $P$ into $n^2$ blocks of $m\times m$ matrices $Q_{i,j}$. Now, transpose each $Q_{i,j}$ (...
- 43.2k
1
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1
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171
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transposing "unrimmed" permutations
Denote the set of $n\times n$ permutation matrices by $\mathfrak{S}_n$. The ordinary transpose preserves this group.
Given $P\in\mathfrak{S}_n$, construct the $n\times n$ matrix ${}^tP$ according to ...
- 43.2k
1
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1
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173
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References request: reflections in coxeter groups
Let $V$ be a vector space. A reflection is a linear map $f: V \to V$ which has an eigenvalue $1$ with multiplicity $n-1$.
Let $S_n$ be the symmetric group on $\{1,\ldots,n\}$. Then the reflections in ...
- 6,201
15
votes
1
answer
1k
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Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$
Is there a bound $B$ such that every 2-generator subgroup
$G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$
whose generators do not satisfy a relation of length $\leq B$ is free?
If it exists, ...
- 19.6k