Timeline for What is the Tutte polynomial encoding?
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| Apr 8, 2014 at 9:41 | comment | added | David E Speyer | We don't know the dimension of the vector space in $K_0$ spanned by all matroids, although we do know from Derksen-Fink that it is spanned by Schubert matroids. Alex and I did some preliminary work investigating positivity properties of this $K$-class, but nothing we are willing to state publicly yet. | |
| Apr 8, 2014 at 9:40 | comment | added | David E Speyer | @მამუკაჯიბლაძე Thanks! Not much more progress beyond what's in our paper, no. (There are results in the paper which are not in this answer.) For the study of the invariant in $K_0(G(d,n))$, the starting point is Derksen-Fink arxiv.org/abs/0908.2988 , which works out the vector space of matroids modulo "obvious $K$-theory relations". This has dimension $\binom{n}{d}$ and has basis the Schubert matroids. Since $\binom{n}{d} = \dim K_0(G(d,n))$, I had hoped the obvious relations were all of them, but they are not, because the fact that $\dim Z \leq n-1$ also gives relations. (continued) | |
| Apr 8, 2014 at 7:39 | comment | added | მამუკა ჯიბლაძე | That's breathtakingly beautiful! Has there been any further progress on this? What about the "best" such invariant, with values in the $K$-theory of the Grassmanian itself? The latter must be known I believe. By analogy with the topological case, it must be somehow related to (co)homology operations... | |
| Nov 3, 2010 at 20:47 | history | answered | David E Speyer | CC BY-SA 2.5 |