All Questions
2,027 questions with no upvoted or accepted answers
4
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157
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Endomorphismrings of maximal submodules.
The question I am interested in answering is the following:
Suppose that for a pair of $d$-dimensional modules $M$ and $N$ over a $k$-algebra ($k$ a field) $R$ we have that $\dim_k \rm{Hom}_R(X,M)\...
- 236
4
votes
0
answers
154
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connectivity in automata by words of length n-1
Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected?
That is for any pair of distinct ...
4
votes
0
answers
525
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Cyclic vector theorem
Could be proved the cyclic vector theorem for a differential module (over a differential field of characteristic zero) by using the fact that the ring of differential operators (over the same ...
- 41
4
votes
0
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570
views
On Grothendieck ring and semiring
We are given a language $L$ and a structure $M$ (model). Definable sets in this model are subsets of $M^n$ definable by a formula of $L$.
The Grothendieck semiring of the structure is defined in the ...
user16974
4
votes
0
answers
1k
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Cartan decomposition for upper triangular matrices
Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...
- 11.2k
4
votes
0
answers
434
views
Smallest matrix covered by many random n by n matrices
We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...
- 51
4
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248
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When are all modules direct factors of a direct product of a fixed one?
Background: For a ring $R$, we denote by ${\rm\mathop Mod}(R)$ the category of all (say right) $R$-modules. If $R$ is pure semisimple, then it is known that ${\rm\mathop Mod}(R)={\rm\mathop Add}(M)$, ...
- 666
4
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3k
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The determinant of the hadamard product of two matrices
We know that the determinant of a Hadamard product of two positive semidefinite matrices $|{\bf A}\circ{\bf B}|$ is greater than or equal to $|{\bf A}||{\bf B}|$. Are there any general results on ...
- 449
4
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261
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Growth of symmetric positive definite integral matrices.
Given an integer $d$, let $\alpha_d(N)$ denote the number of symmetric integral positive definite matrices of size $d\times d$ with coefficients in $\lbrace -N,-N+1,\dots,N-1,N\rbrace$.
...
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4
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answers
257
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A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 18.7k
4
votes
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557
views
Finite Idempotent Semirings (Dioids)
How many finite idempotent semirings (dioids) are there of order n?
And how many have an addition operation that coincides with a maximum operation for some ordering of the elements ?
Even if the ...
- 301
4
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0
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453
views
Convergence of the relaxation method for every parameter in the relevant disk
For large size matrices, the resolution of linear systems $Ax=b$ is often done iteratively. The matrix $A$ is split as $A=M-N$, with $M$ invertible, and one performs
$$x^{k+1}=M^{-1}(Nx^k+b).$$
The ...
- 52.3k
4
votes
0
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562
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The order of the Jacobi method for Hermitian matrices
Let $H$ be an $n\times n$ Hermitian matrix. The Jacobi method is an iterative method for finding the spectrum of $H$. It is described in every book on numerical linear algebra.
Principle: At step $k$,...
- 52.3k
4
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answers
179
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Global dimensions of orders over non-Gorenstein centers
This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
- 30.5k
4
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351
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Combinatorics of signed oriented graphs/skew-symmetric matrices
Consider a "complete" signed graph on $n$ vertices indexed by
$1,2,\dots,n$, that is, a graph in which any two distinct vertices $i$ and $j$ are connected by an oriented edge.
For each pair of ...
- 13.5k
4
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167
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Components of variety of subalgebras
This question is motivated by the question Subalgebras of matrices and its answer by Mariano. We consider $X_{n,d}$, the variety of $d$-dimensional subalgebras not necessarily with 1 (with 1 makes ...
- 12.3k
4
votes
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answers
154
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decomposition of paraunitary matrices
Let $A(X,Y)$ be a matrix with coefficients in $\mathbb{C}[X,X^{-1},Y,Y^{-1}]$, where the conjugation operation is defined on this ring by ${(z X^a Y^b)}^* =\bar{z}X^{-a}Y^{-b}$. So $A$ is said to be ...
- 111
4
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0
answers
733
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Haar measure on strictly upper triangular matrices
Let F be a function field, and A its adele ring. I want to consider U(A)/U(F), where U(A) is the space of strictly upper triangular matrices with entries from A, and U(F) is the same with entries ...
- 41
4
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325
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Localization of power series and module structure
Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable.
Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials.
Let also $\widehat{R}$ be the ring of ...
- 41
4
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answers
534
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Why are Lie Algebras/ Lie Groups so much like crossed modules, and not?
A crossed module consists of a pair of groups $G$ and $H$ with a group homomorphism, $t:H \rightarrow G$, and $\alpha: G \times H \rightarrow H$ that defines an action of $G$ on $H$, $\tilde{\alpha}$: ...
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3
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89
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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
3
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0
answers
155
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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
views
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) : \...
3
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36
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Computing the truncations (“ancestors”) of a surreal number from its Hahn series representation (“normal form”)
If $x$ is a surreal number and $\alpha$ an ordinal, let us denote $T_\alpha(x)$ and call $\alpha$-truncation of $x$ the surreal number whose sign sequence is obtained by truncating the sign sequence ...
- 32.5k
3
votes
0
answers
161
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Generalized dimension property for rings
My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$.
For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
- 223
3
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118
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A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
3
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0
answers
83
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Non-commutative Gorenstein Koszul algebras
I was wondering if there exists a characterization of finite-dimensional Gorenstein Koszul algebras in terms of their Koszul dual?
- 359
3
votes
0
answers
109
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How much a general a theory of matrices equivalence under group actions we have?
Let $F$ be a field and let $M_{m,n}\,(F)$ be the $F$-linear space of $m \times n$ matrices over $F$. Let $G$ be a group acting on $M_{m,n}\,(F)$.
My question is: Do we have some theory about the ...
- 157
3
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0
answers
152
views
My category is rigid: what this implies for representation theory?
I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).
It is clear from our definition of $\mathcal{C}$ that it is ...
- 3,318
3
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0
answers
130
views
Trace map on Ext group
Let $R$ be a (possibly non-commutative) unital ring and $M$ be a perfect left $R$-module. Then, we have the trace map
$$
\operatorname{Tr}\colon \mathrm{Hom}_R(M,M)\to R/[R,R]\,.
$$
According to the ...
- 985
3
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answers
106
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Formulas for the line joining two points in the projective plane over a division algebra
Let $K$ be a[n associative] division algebra (= skew field). By the “projective plane” $\mathbb{P}^2(K)$ over $K$ I mean, as usual, the set of triples $(x,y,z)$ of elements of $K$, not all zero, up ...
- 32.5k
3
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answers
100
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Pairing on a Koszul dual pair
Let $A$ be a graded quadratic algebra over a field $k$, and suppose that it admits the Koszul dual $A^!$. I want to know if there is a natural pairing $A\otimes A^!\to k$ or something similar to this. ...
- 985
3
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0
answers
40
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Filtering a pre-Koszul algebra to get a homogeneous associated graded algebra
In Priddy's paper "Koszul resolutions", on p. 42 he defines an algebra $A$ to be pre-Koszul if it can be written as a quotient of a free algebra $F = F\langle x_i \rangle$ with generators $\{...
- 4,254
3
votes
0
answers
58
views
About a circular variant of Vandermonde matrix
Given an arbitrary $(x_1, \dots, x_n) \in [0, 1]^n$, is there any name/known results for the following $n \times n$ matrix (which is constructed by iterating $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
- 33
3
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0
answers
83
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A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
I apologize if this is too elementary a question, but I have not been able to make much progress.
Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
- 251
3
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137
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Composition of Frobenius $n$-homomorphisms, characteristic-free?
This question is, as so often, a crossbreed of curiosity and laziness. The
former has led me to an interesting, but somewhat unsatisfactory paper by
Khudaverdian and Voronov
(arXiv:2002.02395v2) and, ...
- 33.8k
3
votes
0
answers
161
views
Amalgamation of commutative subrings
Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$.
Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection?
This is equivalent to ask if in the ...
- 39
3
votes
0
answers
202
views
Coevaluation for linear categories
For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...
- 834
3
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answers
134
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Generalized wreath products of commutative algebras with Hopf algebras
Fix $k$ a commutative ring (or, if more convenient, assume it’s a field or even an algebraically closed field of characteristic 0, which is the case I’m mainly interested in). Let $A$ be a unital ...
- 2,830
3
votes
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answers
222
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What is a Gelfand-Tsetlin subalgebra?
For context on general Gelfand-Tsetlin theory, see for instance this MO post.
Let's work over $\mathbb{C}$. Fix $n>0$. There is a natural chain of embeddings of the general linear Lie algebras $\...
- 3,318
3
votes
0
answers
452
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Dimension of a subspace of $n\times n$ real symmetric matrices
Let $n\in \mathbb N.$ Let $W$ be a non-trivial subspace of $n\times n$ symmetric matrices such that for every $x\in \mathbb R^n\setminus \{0\}$ there exists $a_x\in \mathbb R^n\setminus \{0\}$ such ...
- 49
3
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520
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Particular Lie bialgebra structure
Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra, and let $r\in\bigwedge^2\mathfrak{g}$ be a solution of the Yang-Baxter equation. The Yang–Baxter equation states that for all $x\in\...
- 442
3
votes
0
answers
106
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Finite dimensional real division algebra up to isotopy
Finite dimensional real division (non necessarily associative) algebras exist in dimensions 1, 2, 4, and 8. The standard example is a Hurwitz algebra $(A,*)$: reals, complexes, quaternions, octonions. ...
- 12.3k
3
votes
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145
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Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
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3
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answers
83
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How many local maxima can $(x_1,\dots,x_r)\mapsto\|x_1A_1+\dots+x_rA_r\|_\infty/\|(x_1,\dots,x_r)\|_2$ have for Hermitian $A_1,\dots,A_r$?
Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian.
Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting
$$f_{...
3
votes
0
answers
134
views
What is this correspondence between composition algebras over R and superstring theories?
In the page for superstring theory, Wikipedia states:
Another approach to the number of superstring theories refers to the mathematical structure called composition algebra. In the findings of ...
- 131
3
votes
0
answers
130
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The probability that the dominant eigenvalue of a random real matrix is real
Let $X_n$ be an $n\times n$ real matrix where the entries in $X_n$ are independent, normally distributed, have mean $0$, and variance $1$. Suppose that $\lambda_1,\dots,\lambda_n$ are the eigenvalues ...
3
votes
0
answers
225
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Intersection of two modules (and sub-modules) under tensors
I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...
- 31
3
votes
0
answers
85
views
Exterior powers of the Cartan matrix and Dyck paths
(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
- 26.5k
3
votes
0
answers
119
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Finite algebras with finitely many automorphisms
Let $B'/B$ be a finite locally free algebra. Locally in $B$, there is an isomorphism of $B$-modules $B'\simeq B^{\oplus n}$. When is the automorphism group of $B'/B$ finite? When is it unramified? Is ...
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