OFFSET
1,2
COMMENTS
K(x,y) = 2n*Tr(xy)
REFERENCES
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972, 21-22
FORMULA
a(n) = (-1)^binomial(n,2) *2^(n^2-1)*n^(n^2) for n>= 2
MAPLE
interface(rtablesize=infinity):
with(LinearAlgebra):
for n from 1 to 12 do
for i from 1 by 1 to n-1 do
M[i] := Matrix(n);
M[i](i, i) := 1;
M[i](i+1, i+1) := -1;
end do:
ctr := n:
for i from 1 by 1 to n do
for j from 1 by 1 to n do
if(i <> j) then
M[ctr] := Matrix(n);
M[ctr](i, j) := 1;
ctr := ctr +1;
end if
end do:
end do:
A := Matrix(n^2-1):
for i from 1 by 1 to n^2-1 do
for j from 1 by 1 to n^2-1 do
A(i, j) := 2*n*Trace(M[i].M[j]):
end do:
end do:
print(Determinant(A));
end do:
# Alternatively, using the second description
print(0);
for n from 2 to 20 do
print((-1)^(binomial(n, 2))*2^(n^2-1)*n^(n^2));
end do:
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Carmen Bruni, Oct 08 2011
STATUS
approved