Tuesday, 21 February 2012

co.combinatorics - Algebraic Kneser conjecture?

Recall that Kneser conjecture (now Lovasz theorem) claims that if the family of k-subsets (subsets of cardinality k) of given (2k+d)-set M, dgeq1 are colored into d+1 colors, then there are two disjoint k-subsets of the same color (easy to see that for d+2 colors it does not hold, say, if M=1,2,dots,2k+d, then color KsubsetM to color iin1,2,dots,d+1 if minK=i and color K to color d+2 if minK>d+1).



This may be reformulated as follows (mad on first glance) way. Consider the (say, complex) non-commutative algebra A with generators g1, g2, dots, g2k+d and relations gixgi=0 for each i=1,2,dots,2k+d and each xinA. This algebra is naturally graded by degree of monomials in gi's. Denote by Ak homogeneous component of degree k. Note that if Aknix=sumKc(K)g(K) (here K runs over k-subsets of index set 1,2,dots,2k+d, g(K):=gagbgcdots for K=a<b<cdots, C(K) are some complex coefficients), then x2=0 iff no two K's with c(K)ne0 are disjoint. So, the Kneser conjecture is equivalent to the following statement:



not any element Ak is a some of d+1 square roots of 0.



This is nothing but tautology, but the question is whether this statement holds aswell for some factors of G, which have nicer algebraic structure? For example, for commutative algebra with relations g2i=0, or for exterior algebra? (This last may happen only for even k, of course, else each element is a square root of 0).

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