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%I #30 Dec 30 2022 11:31:53
%S 2,1,-6,-2,-16,-9,60,25,576,133,-2016,-1440,-20480,-7008,244944,65745,
%T 2304000,1100736,-20460000,-8997802,-637009920,-142221875,4635933120,
%U 3099895353,108206751744,36079256640,-3045703680000,-709133348000
%N Determinants of the (floor(n/2) - 1) X (floor(n/2) - 1) matrix whose (i,j)-th entry is the intersection number on M_{0,n} of the F-curve F_{1,1,i,n-i-2} and the divisor of the conformal blocks bundle associated to the Lie algebra sl_n, the level 1 and the n-tuple of weights omega_j^n.
%C It is strongly believed that this sequence is never zero. If this were to hold true, these conformal blocks divisors would form a basis of Pic(M_{0,n})^{S_n}.
%D A. Beauville, Conformal blocks, fusion rules and the Verlinde formula, Bar-Ilan University, Ramat Gan, Israël, 1996, pages 75-99.
%D K. Ueno, Conformal field theory with gauge symmetry, American Mathematical Society, 2008
%H A. Beauville, <a href="https://arxiv.org/abs/alg-geom/9405001">Conformal blocks, fusion rules and the Verlinde formula</a>, arXiv:alg-geom/9405001, 1994.
%H N. Fakhruddin, <a href="http://arxiv.org/abs/0904.2918">Chern classes of conformal blocks on M_{0,n}</a>, arxiv:0904.2918 [math.AG], 2009-2011.
%e For n=6 the intersection matrix is [[0,3],[2,0]] giving a determinant of -6. In general these matrices have a high degree of symmetry and periodicity.
%t slncbIntersection[n_, i_, j_] := Module[{k = n-Mod[i j, n]}, If[Floor[i j/ n] + Floor[j (n-i-2)/n] == j-2, If[1 <= k <= j, Return[k], If[j <= k <= 2j-1, Return[2j-k]], Return[0]]]; 0];
%t slnIntersectionMatrix[n_] := Module[{matdict}, Do[r = slncbIntersection[n, i, j]; matdict[i-1, j-2] = r, {i, 1, n/2-1}, {j, 2, n/2}]; Return[ Table[ matdict[i, j], {i, 0, n/2-2}, {j, 0, n/2-2}]]];
%t Table[Det[slnIntersectionMatrix[n]], {n, 4, 31}] (* _Jean-François Alcover_, Aug 10 2018, translated from Sage *)
%o (Macaulay 2 with package ConfBlocks) slnl1intmat = (n) -> ( if even(n) then g=lift(n/2-1,ZZ) else g=lift((n-1)/2-1,ZZ); cu:=sn1curves(n); M = apply(#cu, i -> apply(g, j -> (1/1)*CdotCBslml1(cu_i,n,1,apply(n, k -> j+2)))); matrix M )
%o (Sage)
%o # Based on a formula in "Conformal Blocks Divisors on M_{0,n}"
%o # with Maxim Arap, Angela Gibney and David Swinarski.
%o def sln_cb_intersection(n,i,j):
%o """Gives the intersection on M_{0,n} of the sl_n level 1, omega_j^n conformal blocks divisor with the F-curve F_{1,1,i,n-i - 2}"""
%o k = n - (i*j % n)
%o if (i*j/n).floor() + (j*(n-i-2)/n).floor() == j-2:
%o if k in [1..j]:
%o return k
%o elif k in [j..2*j-1]:
%o return 2*j - k
%o else:
%o return 0
%o def sln_intersection_matrix(n):
%o """gives the g X g intersection matrix of sln level 1 omega_j^n conformal blocks divisors with 1,1,i F-curves"""
%o matdict = dict()
%o for j in [2..(n/2).floor()]:
%o for i in [1..(n/2).floor()-1]:
%o r = sln_cb_intersection(n,i,j)
%o if r != 0:
%o matdict[(i-1,j-2)] = r
%o return matrix(matdict)
%o [sln_intersection_matrix(n).determinant() for n in [4..104]]
%K sign
%O 4,1
%A Jim Stankewicz (stankewicz(AT)gmail.com), Mar 19 2010
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