Questions tagged [tensor-products]

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Certain isotypical component of the tensor product of irreducible representations of $\mathrm{U}(n)$

The following question is closely related to this one. Let $\mathrm{U}(n)$ be the group of all (complex) unitary matrices $n\times n$. It is known that all irreducible representations of $\mathrm{U}(n)...
richrow's user avatar
  • 379
1 vote
0 answers
100 views

Is the rank preserved when the spectral radius is maximized?

If $A$ is a matrix, then let $\rho(A)$ denote the spectral radius of $A$. If $A=(a_{i,j})_{i,j}$, then let $\overline{A}=(\overline{a_{i,j}})_{i,j}$. Suppose that $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ ...
Joseph Van Name's user avatar
5 votes
0 answers
84 views

Recoverability of categorical products in graphs

For simple, undirected graphs $G, H$, let $G\times H$ denote the categorical product, or tensor product, of $G$ and $H$. Let us call a graph $G = (V,E)$ (product-)reducible if there are graphs $G_i = ...
Dominic van der Zypen's user avatar
5 votes
1 answer
283 views

Tensoring with an induced representation: proof question

Let $G$ be a locally compact Hausdorff group and $H$ a closed subgroup of $G$. If $\sigma: H \to B(\mathcal{K}_\sigma)$ is a unitary representation of $G$, we can associate an "induced ...
Andromeda's user avatar
  • 189
3 votes
0 answers
65 views

Tensor product of exact couples

Suppose $(D_1,E_1;i_1,j_1,k_1)$ and $(D_2,E_2;i_2,j_2,k_2)$ are two exact couples, ie., there are exact sequences $D_1\xrightarrow{i_1}D_1\xrightarrow{j_1}E_1\xrightarrow{k_1}D_1$ and $D_2\xrightarrow{...
Faniel's user avatar
  • 623
3 votes
2 answers
657 views

An inequality for the spectral radius of block matrices

Let $d,m$ be positive integers. Suppose that $A_{i,j}$ is a $d\times d$-matrix with real entries whenever $i,j\in\{1,\dots m\}$. Let $A$ be the $dm\times dm$ matrix that can be written as a block ...
Joseph Van Name's user avatar
3 votes
1 answer
229 views

Is $\rho(X_1\dots X_r)^{2/r}\leq \frac{d}{r}\cdot\rho(X_1\otimes X_1+\dots+X_r\otimes X_r)$ for $d\times d$-real matries $X_1,\dots,X_r$?

Let $\rho(A)$ denote the spectral radius of a square matrix $A$. Let $r,d$ be positive integers. Let $X_1,\dots,X_r$ be $d\times d$-real matrices. Then do we necessarily have $$\rho(X_1\dots X_r)^{2/r}...
Joseph Van Name's user avatar
2 votes
0 answers
468 views

Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices

recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
MarcPN's user avatar
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7 votes
1 answer
174 views

Projective tensor product of injective operators

I've seen claims that it is known that for a pair of bounded injective linear operators $T\colon X\to Y, S\colon W\to V$, their tensor product $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...
Tomasz Kania's user avatar
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2 votes
1 answer
149 views

Can the supremum of this quotient of spectral radii be reached?

Let $V$ be a finite dimensional complex inner product space. If $A_1,\dots,A_r\in L(V)$, then define a mapping $\Phi(A_1,\dots,A_r):L(V)\rightarrow L(V)$ by letting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\...
Joseph Van Name's user avatar
2 votes
1 answer
311 views

When does the Cauchy-Schwarz inequality for spectral radii of tensor products become equality?

Let $V$ be a complex finite dimensional inner product space. If $A_{1},\dots,A_{n}:V\rightarrow V$ are linear operators, then let $\Phi(A_{1},\dots,A_{n}):L(V)\rightarrow L(V)$ be the superoperator ...
Joseph Van Name's user avatar
1 vote
1 answer
132 views

The cap set tensor in Lovett (2019)

I hope this is appropriate for the site. I am reading the paper "The analytic rank of a tensor" [S. Lovett, Discrete Analysis (2019), #7, 10 pp.] and am a bit confused in one of the ...
Marcel K. Goh's user avatar
-1 votes
1 answer
238 views

Injection vs. bijection between $V^\star \otimes V^\star$ and $(V \otimes V)^\star$ [closed]

It is a well-known fact that, if $V$ is a vector space over a field $k$, then $V^\star \otimes V^\star$ embeds into $(V \otimes V)^\star$. It turns out to be an isomorphism when $V$ is a finite-...
MathTolliob's user avatar
5 votes
1 answer
161 views

An example where the non-Archimedean tensor product of normed modules is only seminormed?

Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies $$ | 0_R| = ...
dejavu's user avatar
  • 153
11 votes
2 answers
738 views

Tangent bundle of a tensor product bundle

This question was also asked here on math-stackexchange. Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K_E: ...
Raz Kupferman's user avatar
3 votes
0 answers
61 views

Dual space of Carleman functions

Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which $$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...
Janik's user avatar
  • 141
3 votes
1 answer
119 views

Better solution for an evaluation over a fully connected, symmetric tensor network graph?

I have a somewhat nice approach to the following symmetric network evaluation problem. However, since the result is a possibly elaborate combinatorial problem, that I can not yet solve (edit: which ...
Sebastian K.'s user avatar
10 votes
2 answers
1k views

Understanding Balmer spectra

$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated ...
N.B.'s user avatar
  • 757
1 vote
0 answers
73 views

Principal component analysis with boundedness constraints

Let $A$ be an $m\times n$ matrix with entries in $F$ ($F=\mathbb{R}$ or $F=\mathbb{C}$). It is well-known that $A$ has decompositions of the form $$\displaystyle A = \sum_{k=1}^r\lambda_k\hspace{2mm} ...
Onur Oktay's user avatar
  • 2,263
9 votes
3 answers
522 views

Defining the abstract tensor product of W*-algebras via a universal property

I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt: It is easy to show that such an object ...
Andromeda's user avatar
  • 189
2 votes
1 answer
199 views

Need reference for: $\lVert\cdot\rVert_{\text{max}} \leq \lVert\cdot\rVert_h$

Let $A$ and $B$ denote $C^{\ast}$-algebras. Let $\lVert\cdot\rVert_h$ and $\lVert\cdot\rVert_{\text {max}}$ denote the Haagerup norm and max $C^*$-norms on $ A \otimes B$, respectively. I am looking ...
Math Lover's user avatar
  • 1,065
6 votes
1 answer
534 views

What conditions are sufficient for the Leray-Hirsch theorem to be a Künneth formula?

This was originally posted on MSE, and since it didn't receive much attention, I'll try here. Let me know if this is not the appropriate place. Given a fiber bundle $F \to E \to B$ over a paracompact ...
Paul Cusson's user avatar
  • 1,735
5 votes
0 answers
129 views

A concrete description of the projective tensor product of Lipschitz spaces

$\newcommand{\projtenprod}[2]{#1 \; \hat\otimes_\pi #2}$ $\DeclareMathOperator\Lip{Lip}\DeclareMathOperator\AE{AE}$ $\newcommand{\norm}[1]{\| #1\|}$ $\newcommand{\abs}[1]{| #1|}$ Background ...
Yury Korolev's user avatar
11 votes
1 answer
1k views

Are condensed vector spaces over finite fields always solid?

The Clausen-Scholze theory of condensed mathematics offers an abelian category with enough projective objects that embraces the study of arbitrary locally compact (and Hausdorff) groups. The behaviour ...
Peter Kropholler's user avatar
3 votes
1 answer
224 views

Proof of $V\cong \overline{K} \otimes_{K} V_K$ using $H^1(G_{\overline{K}/K},\operatorname{GL}_n(K))=0$

This is from Silverman's book "The arithmetic of elliptic curves" (AEC), p.36, lemma 5.8.1. Lemma 5.8.1 states Let $V$ be a $\overline{K}$-vector space, and assume that $G_{\overline{K}/K}$ ...
Neronoggshafareivh's user avatar
4 votes
1 answer
170 views

If $A\hat\otimes B$ has identity then so are $A$ and $B$

Let $A$ and $B$ be commutative Banach algebra. I have proven that if $A$ and $B$ have identity $e_A$ and $e_B$ respectivly , then $e_A\hat\otimes e_B$ is identity for $A\hat\otimes B$ (the ...
user62498's user avatar
  • 813
-4 votes
1 answer
186 views

Does Rankin-Selberg convolution preserve primitivity?

Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...
Sylvain JULIEN's user avatar
3 votes
1 answer
203 views

Woronowicz characters are multiplicative

I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. Let $G$ be a $C^*$-algebraic compact quantum group with function algebra $(C(G), \Delta)...
Andromeda's user avatar
  • 189
4 votes
0 answers
158 views

Tensor product of representations on a compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$. Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
Andromeda's user avatar
  • 189
4 votes
2 answers
220 views

"Partially strict" monoidal categories

Recall that a monoidal category $\newcommand{\C}{\mathcal{C}} (\C,\otimes,I)$ comes equipped with an associator $\alpha_{XYZ} \colon X \otimes (Y \otimes Z) \xrightarrow{\sim} (X \otimes Y) \otimes Z$ ...
Manny Reyes's user avatar
  • 5,102
3 votes
0 answers
156 views

Tensor product of operator subalgebras and properties of the trace

Note that this question was already posted on MSE: https://math.stackexchange.com/questions/4290741/tensor-product-of-operator-subalgebras-and-properties-of-the-trace Let $V$ be a vector space and let ...
oliverkn's user avatar
  • 139
6 votes
0 answers
148 views

Cech cohomology on product covers & Fréchet sheaves

My question is about the paper [Ka67]. Let $S, T$ be sheaves of nuclear Fréchet spaces over paracompact topological spaces $X, Y$, respectively; in particular, if $V \subset U$ are open subsets in $X$,...
Lukas Miaskiwskyi's user avatar
5 votes
0 answers
75 views

Reference request: Étale base change of differential-graded algebras

I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here. I'm looking for a reference for the following fact, which I've recently stumbled upon: ...
Florian Adler's user avatar
3 votes
3 answers
434 views

On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $

$\DeclareMathOperator\Hom{Hom}$Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\Hom_R(\_,R)$. There is a natural map \begin{align} \Phi_M: M \...
strat's user avatar
  • 291
0 votes
0 answers
116 views

Tensor products and intersections of modules

Is it true that $A\otimes_{\Bbbk} B \cap B \otimes_{\Bbbk} A = B\otimes_{\Bbbk} B$ if $B \subset A$ are $\Bbbk$-modules over a ring $\Bbbk$?. It works for $\Bbbk$ a field. Does it work in any other ...
Gustavo's user avatar
10 votes
1 answer
420 views

Taking the category of sheaves is symmetric monoidal

Let $M$ and $N$ be topological spaces. Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$. It seems to me that the equivalence $$\operatorname{Sh}(M) \...
Chris Kuo's user avatar
  • 525
2 votes
1 answer
225 views

Operation including tensor product or Kronecker product transforming matrix $A$ into matrix $B$

Given two matrices $A$ and $B$: What transformation needs to be applied to transform matrix $A$ into matrix $B$? ...
dtn's user avatar
  • 145
0 votes
0 answers
225 views

Operator norm on tensor product of trace classes is multiplicative

Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
Frederik vom Ende's user avatar
1 vote
0 answers
123 views

Adjacency matrix/tensor operations for graph sequences?

Consider a graph $G=(V,E)$. Its adjacency matrix $A$ is defined by $A_{u,v} = 1$ if $(u,v)\in E$, $0$ otherwise. Consider a vector $x$ that associates a value $x_v$ to each vertex of $G$. Consider the ...
Matthieu Latapy's user avatar
2 votes
1 answer
304 views

Good prime ideals in tensor products of local rings

Let $L/K$ be a field extension. Let $R,S$ be two local commutative $K$-algebras and let $\varphi : R \to S$ be a homomorphism of $K$-algebras, not assumed to be local. Let's call a prime ideal $\...
Martin Brandenburg's user avatar
5 votes
1 answer
401 views

Tensor product of perverse sheaves on flag varieties

I am interested in computing tensor products of perverse sheaves on (partial) flag varieties. For a specific example - consider the product of the big projective on $\mathbb{P}^1$ with itself (This is ...
Adam Gal's user avatar
  • 690
3 votes
0 answers
137 views

What is this SVD (called) with a singular value vector and U and V are tensors?

I am looking for information on a specific type of tensor/matrix decomposition which is quite similar to the SVD for matrices but does not look like the HOSVD since the core tensor is only a vector. ...
user141399's user avatar
5 votes
1 answer
305 views

A tensor category need not be isomorphic to a strict tensor category

This question was originally posted on MSE, but got no answer even after putting a bounty on it, so I'll try here. I'm reading the book "Tensor categories" by Etingof, Gelaki, Nikshych, and ...
Andromeda's user avatar
  • 189
3 votes
0 answers
235 views

Tensor product of associated vector bundles

Let $(P, X, \pi, G)$ and $(P', X, \pi', G')$ be two principal bundles (with Lie groups $G$, $G'$ respectively). Given a vector space $V$ and representations $\rho, \rho'$ of the Lie groups in this ...
José Psicodélico's user avatar
1 vote
1 answer
108 views

The mapping $\iota \otimes S$ on the multiplier algebra

Let $A$ be a non-degenerate algebra with multiplier algebra $M(A)$. Let $S: A \to M(A)$ be an antimultiplicative linear map, i.e. $$S(ab) = S(b)S(a).$$ Consider the mapping $$\iota \otimes S: A \...
Andromeda's user avatar
  • 189
1 vote
1 answer
360 views

Generalization of Sinkhorn’s theorem to stochastic tensors?

Is there an algorithm for mapping a nonnegative tensor / kD array into stochastic form? i.e. assume we have a tensor of unnormalized counts, $c: ℕ^{n^k}$, can we derive a function $f: ℕ^{n^k} → ℝ^{n^k}...
breandan's user avatar
  • 113
4 votes
1 answer
311 views

Tensor product of positive linear maps is positive

Let $\pi_1: A_1 \to B_1$ and $\pi_2: A_2 \to B_2$ be positive linear maps between complex $*$-algebras. Is the mapping $$\pi_1 \otimes \pi_2: A_1 \otimes A_2 \to B_1 \otimes B_2$$ again positive? I.e.,...
Andromeda's user avatar
  • 189
14 votes
1 answer
624 views

Is this "semi-tensor product" something recently invented? Are there other usages of it?

The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
FeedbackLooper's user avatar
8 votes
0 answers
264 views

Generalization of a standard algebraic group theory result for a tensor problem

$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...
Ben's user avatar
  • 1,010
4 votes
0 answers
112 views

Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$

It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
gigi's user avatar
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