Non-Abelian W-representation for GKM

A new kind of anomaly: on W-constraints for GKM

“…However, to make it fully satisfactory, we need a maximally simple origin if this SE -and ME would be just a dream. Unfortunately, as anticipated in [25], the story is not just so simpleand details, though seemingly technical, are quite interesting. The fact that the simplest Ward identity ME is not quite the same as SE, which controls the solution, is an interesting twist of the story and this is what we call anomaly in the title of this paper:…”

“…and, as established recently, 6) Z r has a peculiar non-Abelian W -representation [25,[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], i.e. can be unambiguously described by a single combination of W -constraints, ) Z r possesses character expansion in terms of Hall-Littlewood polynomials [16][17][18][19][20][21][22].…”

“…and the suggestion of [25] to use it as SE -a basic equation which provides Z r as a unique solution in the form of the non-Abelian W -representation. In this paper we call it "the main equation", or just ME to simplify the reference.…”

“…First of all we can substitute the known series for Z 2 {p k } (last time cited in the appendix to [25]),…”

“…No doubt, at technical level these observations are well known to people who worked with GKM, but we make an attempt to summarize them and promote to a more conceptual level. This is needed because of the new accents introduced into the GKM theory quite recently, in [16][17][18][19][20][21][22] and [23]- [25], and the need to explain the origins and the form of the "single equation" [24] and the character expansion in terms of Hall-Littlewood polynomials [16][17][18][19][20][21][22]. We do not achieve these goals in the present paper, but we try to better explain the difficulties of one particularly promising suggestion from [25] -with the hope that it gains attention and will be somehow resolved in the near future.…”