Questions tagged [tensor-powers]
The tensor-powers tag has no usage guidance.
17 questions
1
vote
1
answer
142
views
Diagonalizability, orthogonal diagonalizability of higher order tensors and their being or not being dense in some suitable topology
For our discussion, we'll assume that we're working with $\mathbb{R}^m$ only, but much or all of the following discussion should be carried over immediately to any finite dimensional inner product ...
- 1,465
6
votes
1
answer
195
views
Is there an element in an infinite tensor power of a C*-algebra that is invariant under finite permutations?
Let $A$ be a C*-algebra, and consider the infinite tensor power $A^{\otimes {J}}$, where $J$ is infinite (we consider the minimal or maximal tensor product). To any finite permutation, which is a ...
9
votes
1
answer
477
views
Characterising natural transformations between tensor functors
I would like to know if the following conjecture is correct and if so what's a good citation for its proof.
Let $\mathsf{E}$ be the category of euclidean vector spaces, i.e. objects are finite-...
- 9,650
7
votes
1
answer
294
views
Is every basis for $\bigwedge^kV$ satisfying a "complementary" property a rescaling of a "standard" basis?
This is a cross-post.
Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$
be a basis for $\bigwedge^2V$, where each $\omega_{i_1,i_2}$ is decomposable. Suppose that for every $\...
- 6,741
4
votes
0
answers
274
views
Symmetric power contained in tensor power?
Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be ...
- 2,519
2
votes
0
answers
202
views
The first non-trivial Schur functor [closed]
I am trying to understand the Schur functor $S^{(2,1)}$. Let's try on a vector space $V$ of dimension 3. The general definition is :
$S^{\lambda}V = V^{\otimes n} \otimes_{S_n} V^{\lambda}$
where $V^...
- 891
19
votes
2
answers
1k
views
Exponentiation of vector spaces?
This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...
- 18.4k
3
votes
0
answers
364
views
Symmetric power of an algebra
Given an algebra $A$ over $k$ with characteristic zero and a positive integer $n$, the subspace of $A^{\otimes n}$ consisting of all tensors invariant under the action of all permutations $\sigma\in\...
- 31
0
votes
2
answers
316
views
Tensor powers of an algebra all isomorphic
Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume ...
- 4,985
4
votes
2
answers
448
views
Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers
Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map
$$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...
2
votes
1
answer
123
views
Iterated Reduced Tensor Power of Graded Vector spaces
This might be inappropriate for the MO-level. If so I'll delete it...
Suppose $V$ is a $\mathbb{Z}$-graded vector space and
$\overline{T}(V):=V \oplus V\otimes V \oplus \otimes^3 V \ldots$
is the '...
- 624
6
votes
2
answers
470
views
Alternating multilinear invariants of GL(n) on End (k^n)
Introduction. Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus ...
- 33.8k
6
votes
3
answers
948
views
Torsion-free tensor powers
Let $R$ be an integral domain. If $M$ is an $R$-module such that every tensor power of $M$ over $R$ is $R$-torsion-free, then is $M$ necessarily flat as an $R$-module? If not, then does this ...
- 4,985
1
vote
0
answers
239
views
Decomposition of product of exterior products
Suppose $V$ is a finite dimensional vector space of dimension n.
What is the kernel of the map
$$\bigwedge^p V \otimes \bigwedge^q V ----> \bigwedge^{p+q} V$$ ?
(here $p+q< n$)
Thanks..
...
- 11
4
votes
1
answer
627
views
Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
- 33.8k
2
votes
1
answer
690
views
Restricted universal enveloping algebra of Abelian p-Lie algebra
Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...
- 33.8k
17
votes
5
answers
2k
views
What do gerbes and complex powers of line bundles have to do with each other?
We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but ...
- 44.7k