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# 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.