# Tag Info

Accepted

Accepted

### Recovering a smooth manifold from its tensor fields

Look at $M=S^7$. This has 28 smooth structures, and their tangent bundles are all trivial - compare Parallelizability of the Milnor's exotic spheres in dimension 7 . Thus their tensor algebras do ...
Accepted

### Basis of invariant tensors of rank n in three dimensions

Planar partitions with no singletons works. You need to pick for each $n>1$ some map with certain properties. One way to do this is to just fix a preferred trivalent tree of each size and ...

### Clebsch–Gordan decomposition formula for algebraic groups

Let $G$ denote the group, and suppose one has an enumeration of its irreducible representations $V_{\lambda}$ by some combinatorial objects $\lambda$, like integers for $SU_2$ (or finite dimensional ...
Accepted

### Enriching categories and equivalences

I think most category theorists would answer "yes, obviously", and not bother to write down a proof. But presumably that isn't sufficiently convincing, since you ask the question, so let me try to ...

Accepted

### A different notion of a decomposable symmetric tensor (besides Veronese)

Your $\vee$ is essentially multiplication of polynomials. The variety of tensors $x_1 \vee \dotsb \vee x_m$ corresponds to polynomials that factor as products of linear factors. Points of the (...
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### Origin of the symbol for the tensor product

According to John Aldrich's list of "Earliest Uses of Symbols for Matrices and Vectors", the notation $\times$ for direct product (as well as the name itself) goes back to Wedderburn's 1934 ...

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

The answer is NO, if you count $0$ as a unital Banach algebra. Otherwise, it's YES. Let $e \in A \hat\otimes B$ be a unit and take $b_0\in B$ and $g\in B^*$ with $g(b_0)=1$. Let $g\cdot b_0\in B^*$ be ...
Accepted

### exponential functors on finite dimensional complex vector spaces

In case anyone is still interested in this question: There is a classification of polynomial exponential functors on the category $\mathcal{V}$ of finite-dimensional inner product spaces in terms of ...
You are looking at the vector of coefficients of the polynomial $$\sum_{\epsilon}\prod_{i=1}^{m+2}(1-\epsilon_i x_i)$$ where $\epsilon$ runs over all choices of signs $\pm$ provided there are exactly ...
In general, if $A$ is a field of characteristic $0$ and $B/A, C/A$ are two finite extensions of $A$, then the tensor product $B \otimes_A C$ is isomorphic to the product (i.e. direct product of $K$-...