The identities of symmetric matrices.

Abstract

Let $H\sb{n}$ denote the subspace of symmetric matrices of $M\sb{n},$ the full $n\times n$ matrix algebra with coefficients in a field F. Let $$T\sb{2n}(x\sb1,\...,x\sb{2n-1}; x\sb0)=\sum\limits\sb{\sigma\in S\sb{2n-1}\atop i\equiv 1, 2\ mod\ 4} (-1)\sp{\sigma+i-1}x\sb{\sigma(1)}\cdots \sbsp{x\sb0}{(i)}\cdots x\sb{\sigma(2n-1)},$$and $e(n) = n$ if n is even, n + 1 if n is odd. For all $n\ge 1,\ T\sb{2n}(x\sb1,\...,x\sb{2n-1};\lbrack x\sb{2n},x\sb{2n+1}\rbrack)$ is an identity of $H\sb{n}.$ If $charF \not\vert\ e(n)!,\vert F\vert>2n$ and $n\ne 3,$ then any homogeneous polynomial identity of $H\sb{n}$ of degree 2n + 1 is a consequence of $$T\sb{2n}(x\sb1,\...,x\sb{2n-1}; x\sb{2n}),\ T\sb{2n}(x\sb1,\...,x\sb{2n-1};\lbrack x\sb{2n},x\sb{2n+1}\rbrack)$$by substituting $x\sb{i} \circ x\sb{j}:=x\sb{i}x\sb{j}+x\sb{j}x\sb{i}$ or $x\sb{k}$ for some of their variables or multiplying them by a variable. For n = 3, any identity of $H\sb3$ of degree 7 is a consequence of $$\eqalign{&T\sb6(x\sb1,\...,x\sb5;x\sb6),\ T\sb6(x\sb1,\...,x\sb5;\lbrack x\sb6,x\sb7\rbrack),\cr &Q(x\sb1,\...,x\sb6),\ \lbrack S\sb3(\lbrack x\sb1,x\sb2\rbrack,\lbrack x\sb3,x\sb4\rbrack,\lbrack x\sb5,x\sb6\rbrack),\ x\sb7\rbrack,\cr}$$where $$Q(x\sb1,\...,x\sb6):=\sum\limits\sb{(123),(456)}\ \{\lbrack x\sb1,x\sb2\rbrack\lbrack x\sb3,x\sb4\rbrack\lbrack x\sb5,x\sb6\rbrack\},$$the commutators are the arguments of the triple product $\{abc\}:=abc+cba,$ the sum is taken over cyclic permutations of 1 2 3 and 4 5 6, and $S\sb{n}$ is the standard polynomial. To prove the results, a partial ordering on the homogeneous elements of the free associative algebra F (X) over field F with noncommuting generators $X=\{x\sb1,x\sb2,\...\}$ is defined. Let f be an element of F (X) and n the maximum of the degrees of the variables and the multiplicities of the degrees in f. If f is homogeneous and $charF\ \not\vert\ n!$ then f can be decomposed into a sum of two polynomials $f\sb0$ and $f\sb1$ such that for $0m\le n,\ f\sb0$ is either symmetric or skew-symmetric in all its arguments of degree m according as m is even or odd, and $f\sb1$ is a consequence of polynomials of lower type than f. Osborn's Theorem about the symmetry of the absolutely irreducible polynomial identities is obtained as a corollary. The method we set up here is applicable not only to searching for identities of matrices but also to find the identities of arbitrary algebras.