 # S.V.Duzhin

## Decomposable skew-symmetric functions

A (real) function of $n$ (real) variables is said to be skew-symmetric, if it changes sign whenever any two variables are interchanged:

  f(x_1,...,x_i,...,x_j,...,x_n) = -f(x_1,...,x_j,...,x_i,...x_n).


A skew-symmetric function $f(x_1,...,x_n)$ is decomposable, if there exist functions of one variable $f_1$, ..., $f_n$ such that

  f(x_1,...,x_n) = \det\Vert f_i(x_j)\Vert_{i,j=1}^n.


Problem 1. Find a criterion that a given skew-symmetric function $f(x_1,...,x_n)$ be decomposable.

Theorem 1. In the class of analytic functions (or in any ring of functions without zero divisors) a skew-symmetric function $f(x_1,...,x_n)$ is decomposable if and only if it satisfies the identity \begin{equation}\label{kl}

   f(x_1,x_2,...)f(x_3,x_4,...)
- f(x_1,x_3,...)f(x_2,x_4,...)
+ f(x_1,x_4,...)f(x_2,x_3,...) = 0,    (1)


where the dots mean one and the same set of $(n-2)$ variables. Besides the above notion of (completely) decomposable, one can introduce the notion of partially decomposable skew-symmetric functions. If $\lambda=(\lambda_1,...,\lambda_k)$ is a partition of $n$, then by a $\lambda$-decomposable skew-symmetric function of $n$ variables we understand the complete antisymmetrization of the product of $k$ arbitrary functions of $\lambda_1$, ..., $\lambda_k$ variables. The partition $(1,1,...,1)$ gives completely decomposable functions, while the partition $(n)$ yields the class of all skew-symmetric functions in $n$ variables.

Problem 2. For a given $\lambda$, find a criterion of $\lambda$-decomposability.

For more general classes of functions, the assertion of the above theorem is true only one-way. However, equation (1) is meaningful by itself (it comes from one construction of weight systems in the theory of finite type knot invariants), and the problem, inverse to Problem 1 from the point of view of Theorem 1, presents certain interest.

Problem 3. Find a complete description of the solutions to equation (1) for non-analytic functions $f$. For example, it is interesting to study functions with finite support, or whose support is an algebraic variety in ${\mathbb R}^n$ and the restriction of the function to its support is algebraic. 