6

Let $V$ be a complex vector space of dimension $n$. Denote by $v_1\odot\cdots\odot v_k$ the image of $v_1\otimes\cdots\otimes v_k$ in the symmetric power $\newcommand{\Sym}{\mathrm{Sym}}\Sym^k(V)$. It is well-known that the Elements $v^{\odot k}$ for $v\in V$ generate this space (see, for instance, this answer on math.se), so they must contain a basis.

In other words, let $N=\binom{n+k-1}k$, then there must be $v_1,\ldots,v_N\in V$ with $$\Sym^k V = \mathbb Cv_1^{\odot k} \oplus \cdots \oplus \mathbb C v_N^{\odot k}.$$ I am looking for an explicit description of such a basis. Is such a description known? Is there maybe even a "nice" or somewhat "natural" choice for the $v_i$?

As pointed out by Martin Brandenburg, I certainly allow choosing a basis $x_1,\ldots,x_n\in V$ and expressing the $v_i$ as linear combinations of these.

Community
  • 1
Jesko Hüttenhain
  • 15,116

1 Answers1

4

Years later, I found the book Finite dimensional multilinear algebra (Part 1) by Marvin Marcus. Theorem 1.7 in that book has a nice description:

Let $x_1,\ldots,x_n\in V$ be a basis of $V$. Let $G_{kn}:=\{ (\lambda_1\ge\cdots\ge\lambda_k) \mid \forall i\colon 1\le\lambda_i\le n \}$ be the set of nonincreasing integer sequences of length $k$ with entries ranging from $1$ to $n$. Let $m_t(\lambda):=\#\{ i\mid \lambda_i=t \}$ be the number of times that $t$ appears in the sequence $\lambda$, for $\lambda\in G_{kn}$. Then, define $$ v_\lambda := \sum_{t=1}^n m_t(\lambda)\cdot x_t $$ It is shown in the book that $\{ v_\lambda^{\odot k} \mid \lambda\in G_{kn} \}$ is a basis for $\operatorname{Sym}^k V$.

Jesko Hüttenhain
  • 15,116