All Questions
2,026 questions with no upvoted or accepted answers
5
votes
0
answers
131
views
What is known about this conjectured symmetry in the generalized Radon-Hurwitz numbers?
The generalized Radon-Hurwitz number $\rho(m, n)$ is defined as the
maximal dimension of a subspace contained in $Q_{m,n }$, the subset of all real $m\times n$ matrices $A$ which satisfy $AA^T=\lambda ...
- 13.4k
5
votes
0
answers
154
views
One-parameter family of algebra structures: characterizing trivial deformation as adjoint 2-cocycle being a 2-coboundary
$\DeclareMathOperator\C{\mathbf{C}}$Motivation: this post discusses a simple criterion for a 1-parameter family of $n$-dimensional complex algebras to be a "trivial deformation", i.e., be ...
- 63.9k
5
votes
0
answers
128
views
If power of two matrices becomes equal then stays equal, with left-side row multiplication
During my research on probabilistic automata (in joint Computer Science and Mathematics), I could reduce a certain problem to a problem of matrices.
We are given the (element-wise) non-negative ...
- 51
5
votes
0
answers
104
views
Center of symplectic derivation Lie algebra
Morita–Sakasai–Suzuki studied the graded Lie algebra $\mathfrak{h}_{g,1}$ of symplectic derivations, as well as variations $\mathfrak{h}_{g,\ast}$ and $\mathfrak{h}_g$. This is the Lie algebra of ...
- 637
5
votes
0
answers
139
views
How does a map from permutahedra to associahedra factor through multiplihedra?
Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
- 765
5
votes
0
answers
76
views
Permanent of matrices of finite order
Assume $M$ is a $n \times n$-matrix with entries in $\mathbb{Z}$ such that $M^k$ is the identity matrix for some $k \geq 1$.
Question 1: Is the permanent of $M$ non-zero?
This is tested for many ...
- 26.5k
5
votes
0
answers
79
views
Reference request: associative subalgebras of Cayley algebras are at most 4-dimensional
By a Cayley algebra I mean an 8-dimensional algebra (over an arbitrary field) formed in the Cayley-Dickson process. (They are also called octonion algebras, but I prefer to reserve the term octonion ...
- 2,493
5
votes
0
answers
181
views
Is there a list of all real unital subalgebras of M(2,C)?
Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex ...
- 4,943
5
votes
0
answers
203
views
Number of elements in $\mathrm{GL}(n,p)$ with maximal order
I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.
I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
5
votes
0
answers
248
views
Computing powers of a special matrix fast
I've got an $n\times n$ matrix that is upper-triangular (all zeros below diagonal), diagonal entries are positive, and entries that are not on the diagonal are either $0$ or $1$. An example matrix ...
5
votes
0
answers
345
views
A fusion ring identity
Fusion rings
I'll more or less stick to the presentation given in this question: [1]
We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...
- 309
5
votes
0
answers
96
views
Partitioning the set of Pauli words into abelian pieces
Let $\sigma_x,\sigma_y,\sigma_z$ be the Pauli matrices. A Pauli word of length $n$ is defined as the tensor product $\otimes_{i=1}^n\sigma_i$ of operators $\sigma_1,\dots,\sigma_n\in\{\mathbf 1,\...
- 4,535
5
votes
0
answers
436
views
Non-diagonalizable positive matrices
Let $n\geq 3$ and $E_n$ be the set of $n\times n$ matrices $A$ satisfying the $3$ following properties:
$\bullet$ its entries $(a_{i,j})$ are positive integers.
$\bullet$ the eigenvalues of $A$ are ...
- 3,741
5
votes
0
answers
91
views
elementary matrices over a regular ring
Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...
- 3,472
5
votes
0
answers
164
views
Finiteness questions for enveloping algebras
Let $\mathfrak{g}$ be a Lie algebra over a field of characteristic $0$. The interesting case for these questions is when $\mathfrak{g}$ is infinite dimensional.
Is $U(\mathfrak{g})$ a coherent ...
- 691
5
votes
0
answers
140
views
Open problems about Morita and derived invariants
Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...
- 26.5k
5
votes
0
answers
187
views
Any f.p. faithful simple module over a primitive group ring?
Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$.
There ...
- 1,528
5
votes
0
answers
79
views
Embedding the Mészáros subdivision algebra in an Orlik-Terao localization
The following is an open question (Question 4.1) from my paper $t$-Unique
Reductions for Mészáros's Subdivision Algebra (published version in
SIGMA 2018, and slightly updated preprint
version with ...
- 33.8k
5
votes
0
answers
497
views
Slightly noncommutative Nakayama's lemma?
Nakayama's lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $...
- 4,679
5
votes
0
answers
160
views
Operator norm of a soft thresholded symmetric matrix
Let $A$ be a symmetric real-valued $n\times n$ matrix and let ${\left\|A\right\|_{2\rightarrow 2}} := \max_{\left\|u\right\|_{2}\leq 1} \left\|Au\right\|_{2}$ denotes its operator norm (largest ...
- 159
5
votes
0
answers
86
views
von Neumann regular ring homomorphisms
Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat.
In particular, $\mathrm{id}...
- 858
5
votes
0
answers
296
views
Are these element in a group algebra of a torsion-free group zero divisors?
Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)?
$$1+x+y,\quad 4+x+x^{-1}+y+...
- 2,118
5
votes
0
answers
219
views
Constructing a noncommutative algebra from a commutative algebra
I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
- 111
5
votes
0
answers
113
views
The cyclic modules and self injective ring
It is well-known that if R is a Noetherian ring, and every finitely generated right R-module embeds in projective, then R is a self-injective.
My question is that could one replace "finitely ...
- 107
5
votes
0
answers
193
views
Which rings are the endomorphisms ring of some abelian groups?
Which rings are (isomorphic to) the endomorphisms ring of some abelian group? Is there any necessary and sufficient condition?
- 151
5
votes
0
answers
104
views
Finitely generated submodules of projectives lie inside f. g. projectives?
Let $R$ be a (not necessarily commutative) ring.
If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that
$M \subseteq P'...
- 51
5
votes
0
answers
185
views
$p$-adic valuation in the ring $\mathbb{Z}/p^k\mathbb{Z}$
Assume $p$ is a prime number, $M$ be a non-negative integer and denote by $(\mathbb{Z}/p^M\mathbb{Z})^*$ the units of $\mathbb{Z}/p^M\mathbb{Z}$. Now consider the partition of $\mathbb{Z}/p^M\mathbb{Z}...
- 3,062
5
votes
0
answers
205
views
Resultant of a binomial and a trinomial
Does anyone know of any papers dealing with the resultant of a binomial $x^n+a$ and a trinomial $x^r+bx^s+c$ ?
Even special cases would be of interest.
(The resultant of two binomials is well known.)...
- 81
5
votes
0
answers
1k
views
a new proof of a square matrix is similar to its transpose (not completed)
I know there exist a classical way to prove that a square matrix is similar to its transpose in $C$ using jordan matrices.
I would like to prove it in another way but i have some difficulities to ...
- 159
5
votes
0
answers
352
views
0-1 matrix combinatorial problem
Let $M \in \{0,1\}^{n \times n}$ and let $r_i$ be its $i$-th row. Given constant $p \in (0,1/2]$, let the number of $1$s in each row be at least $p\,n$. Given constant $c \in (0,1)$, what is the ...
- 1,677
5
votes
0
answers
165
views
Real endomorphism algebra of abelian surface is never $\mathbb{C}$?
I'm reading about the Sato Tate conjecture for genus 2, and I came to the paper here. This breaks the conjecture into 6 different parts, based on the real endomorphism algebra of the surface, or the ...
- 171
5
votes
0
answers
166
views
Chains of right annihilators in group rings
See the update below
This problem emanates from a question on not-so-simple random walks on finitely generated groups. But to explain the connections would require an extremely long essay.
Let $G$ be ...
- 4,679
5
votes
0
answers
315
views
Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
- 13.9k
5
votes
0
answers
258
views
What is the relationship between Frobenius extensions and Separable extensions
Let $R\to S$ be an extension of possibly non-commutative rings. I am interested in the relationship between $R\to S$ being Frobenius and it being separable.
If it is a Frobenius extension, then there ...
- 9,650
5
votes
0
answers
99
views
Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,482
5
votes
0
answers
262
views
p-adic analogue of self-adjoint operator
Consider the very well-known result that any Hermitian matrix over $\mathbb{C}$, say $T$, admits a decomposition $T = UDU^*$ where $U$ is unitary and $D$ is diagonal with real entries. I am looking ...
5
votes
0
answers
371
views
free subgroups of $SL_2(\mathbb{Z[i]})$
The group $SL_2(\mathbb{Z})$ contains many free subgroups, for example all of the principal congruence subgroups for $n\geq 3$ and the subgroup $\left\langle \left(\begin{array}{cc}
1 & 2\\
0 &...
- 253
5
votes
0
answers
321
views
Weyl algebra acting on a polynomial ring
Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots,
x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl
algebra. As usual $W$ ...
- 51
5
votes
0
answers
250
views
A question on symmetric functions
Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
- 7,239
5
votes
0
answers
133
views
Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$
Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...
- 1,997
5
votes
0
answers
179
views
When is a Hermitian matrix of the form $g^*g$ for some matrix $g$
I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating.
I'm trying to figure out some properties of ...
- 993
5
votes
0
answers
216
views
Existence or construction of a sequence of orthogonal matrices with three properties
This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...
- 984
5
votes
0
answers
241
views
Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$
I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...
- 356
5
votes
0
answers
187
views
Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$
Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is
\begin{equation}
\Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in S_n}...
- 3,984
5
votes
0
answers
317
views
Hochschild Cohomology of the Quantum Torus
I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
- 201
5
votes
0
answers
376
views
Non-linear positive map
In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...
- 421
5
votes
0
answers
254
views
A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
- 6,206
5
votes
0
answers
620
views
Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
5
votes
0
answers
435
views
When is a Hankel matrix invertible?
I would like to have some conditions that render a general Hankel matrix of the form
\begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} \...
- 327
5
votes
0
answers
290
views
Centralizers of elements in free group algebras
Let $A$ be a group algebra of a free group, and $x \in A$. What is the centralizer of $x$? Is there something like Bergman's theorem for free associative algebras?
