Questions tagged [tensor-powers]

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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 ...
Learning math's user avatar
6 votes
1 answer
180 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 ...
Antonio Lorenzin's user avatar
9 votes
1 answer
440 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-...
Johannes Hahn's user avatar
7 votes
1 answer
289 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 $\...
Asaf Shachar's user avatar
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4 votes
0 answers
244 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 ...
grok's user avatar
  • 2,489
2 votes
0 answers
195 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^...
eti902's user avatar
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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 ...
მამუკა ჯიბლაძე's user avatar
3 votes
0 answers
361 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\...
grffnsn's user avatar
  • 31
0 votes
2 answers
314 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 ...
Jesse Elliott's user avatar
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
117 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 '...
Nevermind's user avatar
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6 votes
2 answers
462 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 ...
darij grinberg's user avatar
6 votes
3 answers
923 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 ...
Jesse Elliott's user avatar
1 vote
0 answers
238 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.. ...
jyoti's user avatar
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4 votes
1 answer
612 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 ...
darij grinberg's user avatar
2 votes
1 answer
678 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 $k$-...
darij grinberg's user avatar
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 ...
Ben Webster's user avatar
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