Physics — and — Combinatorics 2000 potx

Proceedings of the Nagoya 2000 International Workshop Physics and Combinatorics 2000 !£"*» Editors Anatol N. Kirillov and Nadejda Liskova Physics and Combinatorics 2000 [...]... singularities In that case, if we also have CJM//G — G*WMIG, the resolution is said to be crepant If M is a vector space this just means that wM//G — O is trivial 6 Let X C (M//G) x M be the universal family, inherited from the universal family on Hilb | G | (M) We have a G-equivariant commutative diagram X — -• M { P[ M//G (15) —? —> • MjG Bridgeland, King and Reid define a functor $:D(M//G)-*DG(M) (16)... (x) > d has codimension at least 2d— 1 Then M//G is a crepant resolution of singularities of M/G, and the functor $ is an equivalence of categories Let us apply this to our case, M = C 2 " and G = Sn Identifying S H ? The Hilbert scheme Hn is non-singular by a theorem of Fogarty,6 and it is known 13 ' 15 that UJH„ =... polygraph and the Hilbert scheme, and the reason why the former carries geometric information about the latter The trivial bundle OH„ occurs as a direct summand of P, and the natural ample line bundle 0ff n (l) is the highest exterior power of B As special cases of our vanishing theorem we therefore recover the previously known vanishing theorems of Danila 9 for the tautological bundle B and of Kumar and. .. (Taormina, 1994) 12 , Macdonald polynomials and geometry, New perspectives in geometric combinatorics (Billera, Bjorner, Greene, Simion, and Stanley, eds.), MSRI Publications, vol 38, Cambridge University Press, 1999, pp 207-254 13 , Hilbert schemes, polygraphs, and the Macdonald positivity conjecture., Submitted to Journal of the A.M.S (October, 2000) 14 Shrawan Kumar and Jesper Funch Thomsen, Probenius splitting... Here x ranges over the cells in the diagram of the partition /x, and the arm a{x) and leg l(x) denote the number of cells strictly east and north of x, respectively, with the diagram drawn in the French manner with the origin at the southwest corner At I,i we have TiB(I^) = B^(q, t), where ( 53 ) **.(«,*) = J2 ^ Here r and s are the row and column coordinates of a cell in /x, indexed from (0,0) for the... be true The results of this paper and (13) confirm their validity Theorem 4 Conjecture 5.3 is true, and consequently Conjecture 5.1 and Corollary 5.2 are true also Using our preceding results, we can reduce the proof of this theorem to a calculation The Bridgeland-King-Reid isomorphism is equivariant with respect to the T 2 action, so it induces an isomorphism KT (Hn) —> (£2,1) from t h e Grothendieck... where g = 0 and g = 1 respectively denotes the free boson and fermion We call the particle obeying Eq (2.1) "g-on", and study the partition function for the ideal g-on gas For a meanwhile, we suppose that g is a positive integer We set for brevity Vk = e-M^-ri, (2.2) A = e/3w"" 0 -+ A n _i -+ • > A x - + A 0 -> 0 - > • • • (31) quasi-isomorphic to R(n,l) Using the formula for * from Proposition 2.5, we have "9R(n,l) = 0 {—1 ) ® (/>*/* A.) e Moreover, since p is flat, and since the functor -e is a natural direct summand of the identity functor, this . Proceedings of the Nagoya 2000 International Workshop Physics and Combinatorics 2000 !£"*» Editors Anatol N. Kirillov and Nadejda Liskova Physics and Combinatorics 2000