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A196995 Determinant of Killing form K(x,y) of the Lie algebra sl(n,C) for n >=1. 0
0, -128, -5038848, 140737488355328, 5000000000000000000000000, -354400937492545922690672153504784580608, -72317557999158469111384459491956546088110808312359944192, 57896044618658097711785492504343953926634992332820282019728792003956564819968 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..8.

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

Sequence in context: A227661 A016939 A017011 * A214389 A017095 A017191

Adjacent sequences:  A196992 A196993 A196994 * A196996 A196997 A196998

KEYWORD

easy,sign

AUTHOR

Carmen Bruni, Oct 08 2011

STATUS

approved

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Last modified January 25 19:09 EST 2020. Contains 331249 sequences. (Running on oeis4.)