A174421 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.

2, 1, -6, -2, -16, -9, 60, 25, 576, 133, -2016, -1440, -20480, -7008, 244944, 65745, 2304000, 1100736, -20460000, -8997802, -637009920, -142221875, 4635933120, 3099895353, 108206751744, 36079256640, -3045703680000, -709133348000

Offset

4,1

Comments

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}.

References

A. Beauville, Conformal blocks, fusion rules and the Verlinde formula, Bar-Ilan University, Ramat Gan, Israël, 1996, pages 75-99.

K. Ueno, Conformal field theory with gauge symmetry, American Mathematical Society, 2008

Links

Table of n, a(n) for n=4..31.

A. Beauville, Conformal blocks, fusion rules and the Verlinde formula, arXiv:alg-geom/9405001, 1994.

N. Fakhruddin, Chern classes of conformal blocks on M_{0,n}, arxiv:0904.2918 [math.AG], 2009-2011.

Example

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.

Mathematica

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];
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}]]];
Table[Det[slnIntersectionMatrix[n]], {n, 4, 31}] (* Jean-François Alcover, Aug 10 2018, translated from Sage *)

Prog

(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 )
(Sage)
# Based on a formula in "Conformal Blocks Divisors on M_{0, n}"
# with Maxim Arap, Angela Gibney and David Swinarski.
def sln_cb_intersection(n, i, j):
    """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}"""
    k = n - (i*j % n)
    if (i*j/n).floor() + (j*(n-i-2)/n).floor() == j-2:
        if k in [1..j]:
            return k
        elif k in [j..2*j-1]:
            return 2*j - k
    else:
        return 0
def sln_intersection_matrix(n):
    """gives the g X g intersection matrix of sln level 1 omega_j^n conformal blocks divisors with 1, 1, i F-curves"""
    matdict = dict()
    for j in [2..(n/2).floor()]:
        for i in [1..(n/2).floor()-1]:
            r = sln_cb_intersection(n, i, j)
            if r != 0:
                matdict[(i-1, j-2)] = r
    return matrix(matdict)
[sln_intersection_matrix(n).determinant() for n in [4..104]]

Crossrefs

Sequence in context: A239148 A057560 A085592 * A089849 A185330 A217955

Adjacent sequences: A174418 A174419 A174420 * A174422 A174423 A174424

Keyword

sign

Author

Jim Stankewicz (stankewicz(AT)gmail.com), Mar 19 2010

Status

approved