A174421 - OEIS

%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