Polynomial identities for skew-symmetric matrices

Abstract

As mentioned in the introduction, Racine and D'Amour have described all identities for Kn, n < 5. In our research we began at n = 5, and immediately found that 8 is the minimal degree, and that there is a large space of identities for degree 8. More precisely, a space of degree 8 multilinear identities for K5 was computed and it has dimension 1756. A character for this space was also computed and it involves, in its decomposition into irreducible characters, all but 4 of S8's irreducible characters. Our next step was to look into this massive space for "simple" identities: identities that may be written in a closed form. We knew by [R1] that s8 should be present (and we verified that it is), so we next went looking (1.4) for PI's alternating not in all 8 variables (like s8) but alternating in 6 variables and symmetric in 2. We discovered that there is a space of dimension 6 for this type, and we also found a space of dimension 6 for the K4 (PI's alternating in 4 variables, symmetric in 2) and K6 (PI's alternating in 8 variables, symmetric in 2) cases. We found similarities between these last two spaces, and from this we were able to conjecture the family of polynomials for Kn described in 2.1. This conjecture has been verified (via Lemma 1.4.1) for n < 7, and we have proofs (see above, and also [DR]) for n < 5. PI's involving only 2 variables are also "simple", so we looked at those as well. We found that K5 does indeed have PI's of this type, and we were able to first find them (1.4) and then obtain a characterization (2.2). We then repeated these computations for K 6 and found that K 6 does not satisfy any 2-variable identities.