All Questions
6,547 questions
5
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883
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Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
- 1,093
1
vote
0
answers
60
views
Reference for Gröbner-Shirshov algorithm in free restricted Lie algebras
I am searching for a reference on the Gröbner-Shirshov algorithm specifically for free restricted Lie algebras. I have already consulted the textbook by Bokut et al (Gröbner–Shirshov Bases Normal ...
- 1,013
8
votes
1
answer
321
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
1
vote
2
answers
86
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Entrywise $\infty$-norm of squared difference of square roots of matrices
For a positive $n \times n$ definite real matrix $M$ we denote by $\sqrt{M}$ the positive square root of $M$. For an $n \times n$ matrix $A$ denote its entrywise infinity norm by
$$\|A\|_{\infty,\...
- 177
1
vote
0
answers
89
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The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
8
votes
2
answers
596
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If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
5
votes
0
answers
216
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Lifting a morphism between quasi-projective varieties
Let $\mathcal{V}$ be an affine algebraic variety over $\mathbb{R}$, $G$ be a finite group acting freely on $\mathcal{V}$. Consider the quotient space $Y:=\mathcal{V}/G$, which itself is a quasi-...
- 364
15
votes
1
answer
518
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Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
- 1,336
0
votes
0
answers
50
views
Degree of determinant of a (non-monic) matrix polynomial
Let $n=2, 3, \dots$ and consider the matrix polynomial $L(\lambda)=\sum_{k=0}^{\ell}A_k\lambda^k$, where $A_k \in \mathbb{C}^{n\times n}$.
In the so-called monic case (or that can be made monic by ...
- 11
7
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2
answers
487
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Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
0
votes
1
answer
222
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On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 494
2
votes
0
answers
35
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Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 21
2
votes
1
answer
327
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Completion of a local ring is noetherian (under some hypothesis)
I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly:
Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
- 293
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 223
2
votes
0
answers
73
views
Are idempotents in a nonnegatively graded algebra conjugate to homogeneous idempotents?
Let $k$ be a field (assume algebraically closed and $\operatorname{char}(k) = 0$ if it helps) and $R = \bigoplus_{i \geq 0} R_i$ a (unital associative) non-negatively graded $k$-algebra. Furthermore, ...
0
votes
1
answer
102
views
Minimally change matrix with determinant 0
In the following matrix equation, all coefficients $a_{ij}>0$ and all $a_i>0$ and the column sums in the matrix $A$ are all 0
(e.g. $-a_{11}+a_{21}+a_{31}=0$, etc.).
This means that
the ...
5
votes
0
answers
226
views
Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{...
- 985
3
votes
1
answer
108
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Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$
Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial
$$P_\Phi(x) = \sum_{\sigma ...
- 528
1
vote
0
answers
43
views
Inner product on a symmetric sum of tensor products
I have a space $V$ of $6 \times 6$ matrices whose basis is $\{1,S,M\}$; writing the components indices: $\{1_{ij},S_{ij},M_{ij}\}$ where $i,j=1,\cdots, 6$. The inner product on this space is
\begin{...
6
votes
1
answer
315
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Sum of derivative of polynomial over its simple roots
Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form
$$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$
where the sum runs over ...
- 99
3
votes
1
answer
234
views
Is there a matrix that has the completely opposite effect of a Hadamard matrix?
First, let me provide some background on the problem:
In the field of Large Language Model quantization/compressions, outliers (abs of outliers are much larger than the mean of abs of all elements in ...
- 31
0
votes
1
answer
118
views
Configurations of signs in a matrix under certain conditions
I have a combinatorial question which is out of my research area.
Given a $2^k\times 2^k$ matrix $A=[a_{i,j}]$ with entries in $\lbrace0,\pm1\rbrace$, where $k$ is a positive integer. Is it possible ...
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3
votes
2
answers
255
views
Is being graded commutative a necessary condition on $A$ such that $H^*(A)$ is commutative?
If we consider any differential graded algebra $A^\bullet$, then its homology is a graded algebra, since the tensor product interacts well with homology.
A sufficient condidtion for the homology to be ...
- 4,386
2
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1
answer
158
views
The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 91
1
vote
1
answer
122
views
distance in the matrix algebra w.r.t. the nuclear norm
Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
- 375
0
votes
1
answer
127
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Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
- 27
1
vote
0
answers
58
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When is a bimodule that is projective as a right and as a left module also projective as a bimodule
Are there practical criteria for determining when a bimodule that is projective as a right and as a left module is projective as a bimodule? Some illustrative examples of what goes wrong and what goes ...
0
votes
0
answers
68
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
2
votes
1
answer
306
views
Serre functors and global dimensions
Let $k$ be a field.
Let $\mathcal{C}$ be an abelian category (over $k$).
We say that $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that
$$
\operatorname{Ext}^i(M,...
- 889
3
votes
0
answers
89
views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
2
votes
0
answers
71
views
Are the ranks of the following matrices given by these simple expressions?
The question itself is formulated in the title, so below I specify the matrices and expressions mentioned there. In case if this is something known or can be easily deduced from something known, this ...
- 321
0
votes
0
answers
23
views
Existence of a subregular element with abelian centralizer in a quadratic Lie algebra
All Lie algebras here will be finite dimensionnal complex Lie algebra.
We say that such an Lie algebra $\mathfrak{g}$ is quadratic if there exist a skew-symetric, non-degenerate bilinear form or ...
- 188
3
votes
0
answers
154
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Efficient computation of $x^n \bmod g(x)$ in $\mathbb{F}_2[x]$
Related to this question. I wish to compute $x^n \bmod g(x)$ in $\mathbb{F}_2[x]$ for some fixed and known upfront $g$. This problem pops up in computing the 'pure' CRC function of a bit sequence of ...
- 139
3
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0
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117
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Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?
Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
0
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0
answers
59
views
Bimodule endomorphisms of a bimodule over a noncommutative ring
Let $R$ be a noncommutative ring and $M$ an $R$-$R$-bimodule that is projective as left $R$-module. We know that the bimodule of left $R$-module endomorphisms ${}_REnd(M)$ is isomorphic to the tensor ...
1
vote
0
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42
views
Catalog of integral symmetric matrices
Let $g$ be an integral symmetric matrix (perhaps with even diagonal components), and define an equivalence relation $g\sim g'$ if $g=Ug'U^T$ with $U$ a unimodular integral matrix.
For fixed $\det g$ (...
5
votes
1
answer
303
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Efficiently computing $\prod_{i=1}^{n} A_i$
Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix}
0 & 1\\
i^k & 1
\end{bmatrix}?$$
I know if $k=0$, we can use ...
- 696
3
votes
1
answer
187
views
Eigenvalues of certain matrices
We write $R(\theta)=\left(\begin{smallmatrix}\cos(2\pi\theta)&\sin(2\pi\theta)\\ -\sin(2\pi\theta)&\cos(2\pi\theta)\end{smallmatrix}\right)$ for any $\theta\in\mathbb R$.
Let $d,m,n,r$ be a ...
- 2,446
8
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1
answer
685
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The state of the art on topological rings - the Jacobson topology
I was recently studying the Jacobson density theorem and I found it quite interesting.
Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it ...
- 183
15
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0
answers
217
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If a map between unital rings preserves multiplication and successor, does it preserve addition?
Welcome to my first MathOverflow posting!
This is a question about rings, all of them assumed to be both unital and associative.
Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
- 151
5
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1
answer
367
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Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.7k
1
vote
1
answer
52
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Eigendecomposition of a Gram matrix with specified columns
I originally posted this question on Math SE, but I am starting to think it is more suitable to post it here. After waiting about two weeks, I didn't get any activity. The original question is linked ...
- 111
2
votes
0
answers
67
views
Nonlinear random matrix equations
Let $C$ be a matrix; $v$ be a column vector;
$P$, $\Delta$ are random matrices;
$x$ is a random column vector.
$$Cvv^T - \mathbb{E}[P^Tx]v^T - \mathbb{E}[P^Tx v^T \Delta] + O(\Delta^2)= 0$$
$$C^TCv - ...
- 21
0
votes
0
answers
52
views
What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?
How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
- 109
4
votes
2
answers
227
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Arithmetic application: Complete group ring and group ring for infinite group
Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\...
- 319
1
vote
0
answers
63
views
The rank of a matrix expression
I'm studying discrete-time LTI systems and state estimators for them. Recently, I studied this paper. I am facing a matrix rank calculation problem and having trouble solving it. I will provide more ...
- 4,484
0
votes
0
answers
53
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Relations between the optimal solutions of two related SDPs
In system theory, we often encounter Semi-Definite Programs (SDPs) with Linear Matrix Inequality (LMI) constraints, such as those presented in this paper. I have introduced new variables based on the ...
- 4,484
1
vote
1
answer
112
views
Coprime polynomials and polynomial substitution
Let $F$ be a field, and let $P(X_1,\dots,X_m)$, $Q(X_1,\dots,X_m) \in F[X_1,\dots,X_m]$ be two coprime polynomials. Consider $n$ new polynomials $R_1(Y_{1,1},\dots,Y_{1,n}) \in F[Y_{1,1},\dots,Y_{1,n}]...
- 640
1
vote
0
answers
121
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Simple algorithm for A107670
Let $T(n, k)$ be A107670 (i.e., matrix square of triangle A107667). Here we define the triangular matrix $P$ by $P(n, k) = \frac{(n+1)^{2(n-k)}}{(n-k)!}$ for $0 \leqslant k \leqslant n$ and the ...
- 4,928