Skip to main content

All Questions

2,027 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
5 votes
0 answers
327 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
user1189053's user avatar
5 votes
0 answers
255 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot \lambda^\...
ScienceSnake's user avatar
5 votes
0 answers
245 views

Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$. Q1) Is it ...
Subhajit Jana's user avatar
5 votes
0 answers
270 views

A generalization of real characters on a group

Yesterday I understood that I can't live without this construction: Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps $\varphi_k:...
Sergei Akbarov's user avatar
5 votes
0 answers
428 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
yang's user avatar
  • 181
5 votes
0 answers
220 views

Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients. $$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$ Monomials $x^k$ are mapped to $n ...
r_faszanatas's user avatar
5 votes
0 answers
2k views

A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: $$Tr\left(\frac{1}{1-AA^T}\right)...
math110's user avatar
  • 4,280
5 votes
0 answers
148 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
Nathaniel Johnston's user avatar
5 votes
0 answers
195 views

Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...
Fred.Fred's user avatar
  • 409
5 votes
0 answers
843 views

Chinese remainder theorem

For non-commutative rings, we have this generalization of the Chinese remainder theorem (CRT). I wonder if there is another statement involving only left or right ideals; do you know any?
Exodd's user avatar
  • 59
5 votes
0 answers
231 views

Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity

Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for $k=0,\dots,l-1$....
Binzhou Xia's user avatar
5 votes
0 answers
392 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
Jeff's user avatar
  • 500
5 votes
0 answers
112 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. $(A_1+A_2)^{-1},(A_1+A_3)^{-1},(A_2+A_3)...
Tzonathan's user avatar
5 votes
0 answers
246 views

Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $\mathbf{k}$ be a commutative ring. Let $C$ be a filtered $\mathbf{k}$-coalgebra. This means a $\mathbf{k}$-coalgebra equipped with an increasing $\mathbf{k}$-module filtration $C^0 \subseteq C^1 \...
darij grinberg's user avatar
5 votes
0 answers
153 views

What is the composition in SesquiAlg?

To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...
Theo Johnson-Freyd's user avatar
5 votes
0 answers
406 views

On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix. Suppose there exists ...
gondolf's user avatar
  • 1,503
5 votes
0 answers
426 views

Homological dimension of completed path algebras.

Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations. Is it true that the I-adic completion of A has finite homological dimension?
Andre 's user avatar
  • 51
5 votes
0 answers
272 views

Cocontinuous monadic functors

Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an ...
Martin Brandenburg's user avatar
5 votes
0 answers
160 views

Is a *complete ring complete in the graded category ?

The question concerns a definition of Bruns, Herzog: Cohen-Macaulay Rings (before Prop. 3.6.16): The Noetherian *local ring $(R,m)$ is said to be *complete if $(R_0,m_0)$ is complete. (a graded ...
Ralph's user avatar
  • 16.2k
5 votes
0 answers
629 views

Elementary polynomial-free proofs of fundamental theorem of Galois theory?

I am looking for simple proofs that show the correspondence between intermediate fields in a field extension and subgroups of the Galois group. I'm happy for everything to be subfields of $\mathbb{C}$....
TLss's user avatar
  • 151
5 votes
0 answers
442 views

A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings? In fact this question is a ...
Alexander Shamov's user avatar
5 votes
0 answers
183 views

integral versus adjoint action on Hopf algebra

Suppose that $H$ is a finite dimensional Hopf algebra (with counit $\varepsilon$) and $T$ is a non zero right integral of $H^{\star}$ (the dual Hopf algebra). Let $ad_h$ be the adjoint action on $H$, ...
user25332's user avatar
5 votes
0 answers
241 views

Roots of unity in algebraic K-theory

For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit. For $...
Craig Westerland's user avatar
5 votes
0 answers
115 views

Ring properties which can be checked on sufficiently many Ore localizations

Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of ...
Greg Muller's user avatar
5 votes
0 answers
1k views

Characteristic polynomial of a symmetric integer matrix

I am wondering what, if anything, is known about the characteristic polynomials of integer symmetric matrices. I believe I read somewhere that not every polynomial with integer coefficients can be a ...
Elena's user avatar
  • 315
5 votes
0 answers
824 views

Unitary unit conjecture for group rings

The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g....
Joerg Sixt's user avatar
5 votes
0 answers
3k views

Matrix normalization for constant row and column sums

From a mxn matrix M with only strictly positive elements, I want to multiply each row and each column by real values in order to obtain a matrix N whose row sums are all 1/m and column sums are all 1/...
Arnaud's user avatar
  • 51
5 votes
0 answers
517 views

Monomial-type ideals in polynomial rings

Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
Timothy Wagner's user avatar
5 votes
0 answers
510 views

localization at central maximal ideals

If M is a maximal ideal of Z(R), the center of a ring with identity R, and R_M, the localization of R at M, is a commutative field, what can we say about R? My guess is that we can's say that much ...
iravan's user avatar
  • 59
5 votes
0 answers
445 views

symplectic matrices

If $( A B | C D)$ is a symplectic matrix with entries in the finite field with two elements, is it necessarily the case that $\sum_{i,j,k} a_{ij}b_{ij}c_{ik}d_{ik} = 0$? This arose in connection ...
Robert Gunning's user avatar
5 votes
0 answers
388 views

is there a notion of weakly noetherian?

A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left ...
Carl Weisman's user avatar
5 votes
0 answers
519 views

Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups

In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...
Mikola's user avatar
  • 2,392
5 votes
0 answers
539 views

An inverse eigenvalue problem on Jacobi matrices

I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
AlastairK's user avatar
  • 291
5 votes
1 answer
226 views

Examples of noncommutative Bezout domains

I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...
4 votes
0 answers
101 views

Dimension of the intersection of the commuting variety with a particular subspace

Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as: $$ \mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}. $$ It is well known that $\...
darko's user avatar
  • 309
4 votes
0 answers
168 views

How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?

I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
tongjun's user avatar
  • 41
4 votes
0 answers
132 views

Ring theoretical aspects of the DAHA

The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively). Nowdays there are many variations of the ...
jg1896's user avatar
  • 3,318
4 votes
0 answers
211 views

When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?

$\hspace{20pt}$Duplicate on stackexchange. This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
cnikbesku's user avatar
  • 171
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 \...
user237522's user avatar
  • 2,837
4 votes
0 answers
158 views

Wedderburn-Malcev principal theorem for graded-finite algebras

Let $k$ be a field and $A$ be a noncommutative $k$-algebra with Jacobson radical $J$. If $A$ is finite-dimensional, the Wedderburn-Malcev says that $A$ has a subalgebra $S$ such that $$A = S \oplus J$$...
Alvaro Martinez's user avatar
4 votes
0 answers
79 views

Closed character formula for the module $L(a\omega_i)$

Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\omega_1, \omega_2, \dots, \omega_n\in\mathfrak{h}^{*}$ is the ...
user1324474's user avatar
4 votes
0 answers
99 views

If matrices $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup \mathbb{T}$, does $\text{rank}([A,B])=\text{rank}([A^*,B])$?

Here $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$ denotes the unit circle in the complex numbers. This holds, if we have $\sigma(A)\subseteq \mathbb{R}$ or $\sigma(A)\subseteq \mathbb{T}$ (independent of $B$...
mathemagician99's user avatar
4 votes
0 answers
116 views

When is the intersection of cosets of a conjugacy class $0$-dimensional?

Let $G = \mathrm{SL}_n$ (say); let $K$ be a field. Let $g$ be a regular semisimple element of $G(K)$, and $\mathrm{Cl}_g$ its conjugacy class, considered as an algebraic variety. Then $\mathrm{Cl}_g$ ...
H A Helfgott's user avatar
  • 20.2k
4 votes
0 answers
119 views

Adjoining new factors for primes in UFDs

It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
Pace Nielsen's user avatar
  • 18.7k
4 votes
0 answers
453 views

Problem 1.8 from Kirby's list

Context I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
saver_of_light's user avatar
4 votes
1 answer
363 views

Is there a "natural" interpretation of the power function for octonions and for sedenions?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it is impossible to define $\log(x)$ ...
Dieter Kadelka's user avatar
4 votes
0 answers
326 views

Are there infinitely many simple integral fusion rings of rank $4$?

$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
Sebastien Palcoux's user avatar
4 votes
0 answers
143 views

On the conditions for Artin-Schelter Gorenstein algebras

Let $ k $ be a field and $ A $ a connected graded $ k $-algebra ($ A $ is associative, but not assumed to be commutative). The algebra $ A $ is called Artin-Schelter Gorenstein* of dimension $ d $ if ...
Cranium Clamp's user avatar
4 votes
0 answers
262 views

Two questions about three circulant matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$ $$2AA^T+BB^T+CC^T=(4n+4)I-4J$$ where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
user369335's user avatar
4 votes
0 answers
93 views

Vershik's conjecture about generic quadratic algebras

Is it still unknown whether very general (lying in a countable intersection of some Zariski opens in corresponding Grassmannian) quadratic algebras $R$ with $\operatorname{dim} R_2 < \frac{3}{4}(\...
Denis T's user avatar
  • 4,600

1
6 7
8
9 10
41