Questions tagged [tensor-products]
The tensor-products tag has no summary.
446 questions
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K-flat complexes and unbounded derived tensor product over a non-commutative sheaf of rings
$\def\R{\mathscr{R}}
\def\O{\mathcal{O}}$Let $X$ be a topological space and let $\R$ be a sheaf of unital non-commutative rings over $X$.
When $\R$ is commutative, there is much literature on ...
32
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1
answer
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How large can subspaces of $U \otimes V$ be that avoid any pure tensors?
The Question is simple, yet I have encountered it multiple times in my mathematical life without finding an obvious answer, so I've decided to post it here.
Say $U, V$ are real vector spaces of ...
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0
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Existence of irrational eigenvalues of a sum of representation matrices
Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$. It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $...
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Functional-analytic and Banach space approach to spaces of finite signed measures on $\mathbb{R}$ and $\mathbb{R}^d$
I am looking for good references (survey, monograph, or paper with a solid background section) on the Banach space / functional analytic structure of spaces of finite signed measures $\mathcal{M}(X)$, ...
6
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2
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167
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Dimension of the $H$ fixed subspace of tensor product of representations
Let $G$ be a finite group and $H$ be a normal subgroup of $G$ of index $2$. Let $\operatorname{IRR}(G)$ denote the set of all inequivalent irreducible representations of $G$. For any representation $(\...
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2
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1
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Eigenvalues of a sum of tensor product of representation matrices
Let $G$ be a finite group and $H\le G$. Let $\lbrace{\rho_1,\rho_2,\ldots,\rho_t}\rbrace$ be the set of inequivalent, irreducible representations of $G$, where $\rho_1$ is the trivial representation ...
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6
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Is $\text{Hom}_\mathcal C(x,a)\otimes_{\text{End}_\mathcal C(a)} \text{Hom}_\mathcal C(a,y) \to \text{Hom}_\mathcal C(x,y) $ injective?
[Cross-posted from math.stackexchange (I hope not to insult this forum).]
In words, is the relative tensor product of hom-spaces isomorphic to the range of their composition map?
One could define a ...
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A higher dimensional version of the Bruhat decomposition
The classical Bruhat decomposition states that every invertible matrix $M$ over a field $K$ can be written as $M= U_1 P U_2$ with $U_i$ upper triangular matrices and $P$ a permutation matrix. We can ...
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Is the map of sheaves $\mathcal{F}\otimes \prod_{\mathbb{N}}\underline{\mathbb{Z}}\to \prod_{\mathbb{N}} \mathcal{F}$ always injective?
For an Abelian group $M$ and a collection of Abelian groups $(N_i)_{i\in I}$, the natural map $M\otimes\prod_{i\in I}N_i\to \prod_{i\in I}M\otimes N_i$ can fail to be injective (see here). However, it ...
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Categorical interpretation of the injective tensor product of Banach spaces
The completed projective tensor product $X\hat\otimes_\pi E$ of Banach spaces is well known to be left adjoint to the Hom-functor $L(E,X)$ of continuous (or contractive) linear maps, more precisely, ...
6
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Projective tensor product vs Condensed tensor product
For （compactly generated）topological abelian groups X and Y， there is （uncompleted） projective tensor product $X\otimes Y$ which is the finest topology making $X\times Y\rightarrow X\otimes Y$ jointly ...
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Relation between local-to-global principle in stratification and topology of the Balmer spectrum
The context of my question is the theory of stratification of a tensor triangulated category via Balmer-Favi support recently developed by Barthel, Heard and Sanders in their paper "...
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A K-flat complex is acyclic for the pullback functor. Does the converse hold?
$\def\F{\mathscr{F}}
\def\O{\mathscr{O}}
\def\G{\mathscr{G}}
\def\H{\mathscr{H}}$Let $X$ be a ringed space, and let $\F\in K(X):=K(\O_X\text{-Mod})$ be a complex of $\O_X$-modules. If $\F$ is K-flat, ...
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An almost sure central limit theorem applies to the solution to SPDE
Recently, I read a article named Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator. During the proof of an almost sure ...
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2
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Associativity of projective tensor product
Let $X$ and $Y$ be Hausdorff topological vector spaces. The projective topology is the strongest topology on algebraic tensor space $X\otimes Y$ such that the canonical map $X\times Y\to X\otimes Y$ ...
- 1,511
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76
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vector membership and tensor product
Given a $m$-dimensional nonzero complex vector $\bar{x}$ and a set $S=\{\bar{y}_1,\cdots,\bar{y_n}\}$,for any $k$,
$$
\bar{x}^{\otimes k}\in span\{\bar{y}_1^{\otimes k},\cdots,\bar{y_n}^{\otimes k}\}
$...
- 1,533
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1
answer
161
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Can $\|\cdot\|_{L^p}$ be a uniform cross-norm on the category of Banach spaces?
I am using the formulation from Introduction to Tensor Products of Banach Spaces by Raymond Ryan.
A uniform cross-norm $\alpha$ is an assignment of a pair of Banach spaces $X$ and $Y$ to a norm $\|\...
- 1,511
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Why in mathematical texts the relative order of the top and the bottom tensorial indices is rarely considered?
I decided to repost this question from MSE to here after almost a year without an answer, and because now I need to explain this topic to someone else, while still not completely understanding it.
...
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2
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60
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Explicit chain map for coproduct in cyclic homology
Let $A$ and $A'$ be (unital) algebras over a field $k$. Then there is a "coproduct" map in cyclic homology,
$$HC_n(A \otimes A') \longrightarrow \bigoplus_{i+j=n} H_i(A) \otimes H_j(A'),$$
...
- 10.2k
2
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1
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369
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Best notation for tensor product with associativity
I want to express the product of a three-dimensional array by two one-dimensional vectors over some ring $R$:
$$r = A \cdot b \otimes c$$
where
$A \in R^{\ell \times m \times n}$
$b \in R^n$
$c \in R^...
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1
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1
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127
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Expression of $\partial\left(A^{-1} \times B\right)/\partial Y$
In order to enhance the performances of my ODE solver, I want to provide it the Jacobian matrix of my system which writes:
$$\frac{dY}{dt} = A^{-1}\left(Y\right)\times B\left(Y\right)$$
with $A$ a $N\...
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2
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1
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302
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Question on bitensors
I apologize in advance if this question seems too basic, but I am struggeling to find any references for what appears to be quite simple.
In various contexts, especially in general relativity and ...
user199422
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1
answer
352
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Infinite tensor product of $L^\infty(0,1)$
I was reading Theory of Operator Algebras III by Masamichi Takesaki, specifically the section on the infinite tensor product of von Neumann algebras, and I encountered a question that I would ...
user532538
6
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1
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320
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Classifying associators for the tensor product of graded modules
$\newcommand{\Z}{\mathbb{Z}}$Let $\mathsf{Gr}_{\mathbb{Z}}\mathsf{Mod}_R$ be the category of $\Z$-graded $R$-modules, equipped with the usual tensor product $\otimes$. Using the pentagon condition for ...
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0
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120
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primitive idempotents in semisimple group algebras
Let $G$ be a finite group, and $M$ be a minimal left ideal of $\mathbb{R}G$ (or irreducible $\mathbb{R}$-representation of $G$).
There are three possibilities for $M$:
Case 1: $M \otimes \mathbb{C}$ ...
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4
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1
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$\rm{Tor}$ sheaf is trivial if two sheaves have disjoint support
We know that the support of tensor product of two sheaves $\mathcal{F}\otimes \mathcal{G}$ on a smooth projective variety is just $\text{Supp}(\mathcal{F})\cap \text{Supp}(\mathcal{G})$. Then what can ...
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0
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107
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Nonnegativity of a Hermitian form on orthonormal product states
Let $n \ge 3$ and let $\{e_1,\ldots,e_n\}$ be the standard basis of $\mathbb{C}^n$. Define a Hermitian operator $H \in \textrm{End}\big(\bigotimes_{i=1}^n \mathbb{C}^n\big)$
by prescribing its matrix ...
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1
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567
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Reference for basic multicategory theory
A multicategory (or coloured operad, if you like) is a category where instead of (only) morphisms between objects we have multimorphisms from finite lists of objects to (single) objects.
There may be ...
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Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
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4
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1
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156
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Tight upper bound on a ratio involving symmetric PSD matrices and Kronecker products
Let $\mathbf{A}_i \in \mathbb{R}^{d \times d}$ ($i = 1, \dots, T$) be symmetric positive semidefinite (PSD) matrices. Define the quantity
$$
m = \frac{\lambda_{\max}\left(\sum_{i=1}^T \mathbf{A}_i^2\...
- 123
3
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0
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127
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Commutator of $A\otimes I$ and $I \otimes B$ vanishes?
Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
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1
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Is there a relative projective tensor (cross-)norm for Banach $A$-algebras?
$\newcommand\norm[1]{\lVert#1\rVert}$I am interested in a relative version of the projective tensor product and projective tensor (cross-)norm for Banach algebras. Let $A$, $B$, $C$ be commutative (...
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0
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An application to product formula of multiple integral
It is well-known that from Nualart's book, two multiple integrals can be expanded into a sum of multiple integrals, i.e.,
$$I_n(f)I_m(g)=\sum_{i=0}^{m\wedge n}i!C_m^iC_n^iI_{m+n-2i}(f\otimes_ig),$$
...
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112
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Monoidal categories with canonical left-strengths of monads
It is well-known that every monad on Set is left-strong and that left-strength on a Set-monad is unique.
Is there an abstract characterization of monoidal categories $C$ for which every monad on $C$ ...
- 1,357
1
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0
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75
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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{...
5
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1
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217
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Does the Okounkov-Vershik approach to the representation theory of $S_n$ shed new light on the problem of computing Kronecker coefficients?
I am studying the problem of decomposing tensor products of irreps of $S_n$. As a non-expert, I was surprised to see that this is an open problem, or a the very least, one for which no satisfactory ...
7
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1
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359
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Norm in the minimal tensor product of C*-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
- 3,537
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0
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When does an inner product on $C(K)$ come from integration?
Let $K$ be a compact Hausdorff space and suppose $\langle \cdot, \cdot \rangle$ is an inner product on $C(K)$ such that $\langle f, g \rangle \ge 0$ whenever $f(t),g(t)\ge 0$ for all $t \in K$. It ...
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The Balmer spectrum and the thick tensor ideals of the derived category of a Hopf algebra
Given a Hopf algebra $H$ over a field $\mathbb{k}$, the category of finite-dimensional left-$H$-modules naturally becomes a rigid monoidal category with exact monoidal product. Thus clearly the ...
- 1,584
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1
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874
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Can we test if an abelian group is finitely generated by taking tensor product?
If $A$ is a finitely generated abelian group, then we know that for all fields $k$, the tensor product $A\otimes_{\mathbb{Z}}k$ is finite dimensional as a $k$-vector space.
The converse is not true, ...
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2
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543
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Irreducible tensor product representations in finite simple groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:
A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...
- 27.6k
2
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0
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166
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Tensor product of finite extensions of $\mathbb{Q}_p$
Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.)
$(1)$ If $M$ is a finite Galois extension of $F$ with Galois ...
- 632
1
vote
0
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178
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Vanishing (infinite) tensor products
Since the advent of free probabilities and QFT, infinite tensor products of $R$-associative algebras with units has become more familiar to the working mathematician.
Starting from the (permuting) ...
- 3,786
2
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1
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256
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Combination of simple tensors - II
This is a follow-up question to Combination of simple tensors.
I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two ...
3
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0
answers
233
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Multiplicative structure on Čech–Alexander complexes
I have the following basic question on Čech–Alexander complexes.
Let $R$ be a ring and $A$ be an $R$-algebra. To this datum one can attach a cosimplicial ring which assigns to an object $[n]=\{0,1,\...
- 1,573
1
vote
1
answer
226
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Carnot–Carathéodory norm and the inner product norm
It is well-known that given the extended tensor algebra $T((\mathbb{R}^d))$ one may extract a separable Hilbert space by considering the subset
$$T^1((\mathbb{R}^d)) := \left\{h \in T((\mathbb{R}^d)) :...
- 181
2
votes
1
answer
281
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Combination of simple tensors
I aksed this question on Math Stack Exchange 6 days ago, with no answer: https://math.stackexchange.com/q/4875445/1297919
Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. ...
3
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1
answer
189
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Is there any (specially Algebraic Geometrical) exposition of Koike Terada's Young-diagrammatic methods for the representation theory paper?
I am talking about the paper by Koike, Kazuhiko and Terada, Itaru, Young-diagrammatic methods for the representation theory of the classical groups of type ($B_n$), ($C_n$), ($D_n$), J. Algebra 107, ...
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1
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0
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67
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Inner product of signatures of piecewise linear paths
It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$...
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2
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1
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308
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Are projective tensor products left-exact if one considers only maps of norm at most 1?
Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...
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