Determining the structure of the Schofield resolution of a preprojective algebra. Nothing interesting happens, just extracting information from the Cartan matrix.

cartan.py

def mA(n):
  if n % 2 == 1: return (n - 1) / 2
  else:          return (n - 2) / 2

def mD(n):
  if n % 2 == 1: return (n - 3) / 2
  else:          return (n - 2) / 2

def u(n):
  if n % 2 == 1: return mD(n) + 1
  else:          return mD(n)

# calculate the Cartan matrix for P(T_n)
def cartan(T, n):
  assert T in ["A", "D", "E", "L"]
  assert n >= 1

  C = matrix(n)

  if T == "A":
    for i in range(mA(n) + 1):
      for j in range(n):
        if j <= i:           C[i, j] = j + 1
        elif j >= n - i - 1: C[i, j] = n - j
        else:                C[i, j] = i + 1

    for i in range(mA(n) + 1, n):
      C[i] = C[n - i - 1]

  elif T == "D":
    assert n >= 4
    C[0, 0] = C[1, 1] = mD(n) + 1
    C[1, 0] = C[0, 1] = u(n)

    for i in range(1, n - 1):
      for j in range(n - i - 1):
        C[n - i, n - i - j] = C[n - i - j, n - i] = 2*i

    for i in range(2, n):
      for j in range(2):
        C[i, j] = C[j, i] = n - i
  
  elif T == "E":
    assert n == 6
    if n == 6:
      C = matrix([
        [4, 2, 4, 6, 4, 2],
        [2, 2, 3, 4, 3, 2],
        [4, 3, 6, 8, 6, 3],
        [6, 4, 8, 12, 8, 4],
        [4, 3, 6, 8, 6, 3],
        [2, 2, 3, 4, 3, 2]])

  elif T == "L":
    for i in range(1, n + 1):
      for j in range(n - i + 1):
        C[n - i, n - i - j] = C[n - i - j, n - i] = 2*i
  
  return C

# calculate the dimensions of the projective resolution of P(T_n)
def resolution(T, n):
  C = cartan(T, n)
  R = [0, 0, 0, 0, 0, 0, 0]

  R[1] = R[5] = sum(sum(C))
  R[2] = R[4] = sum([sum(C[i]) ** 2 for i in range(n)])

  if T == "A":
    R[3] = sum([2 * sum(C[i]) * sum(C[i + 1]) for i in range(n - 1)])
  if T == "D":
    R[3] = 2 * sum(C[0]) * sum(C[2]) + 2 * sum(C[1]) * sum(C[2]) + sum([2 * sum(C[i]) * sum(C[i + 1]) for i in range(2, n - 1)])
  if T == "E":
    if n == 6:
      R[3] = 2 * sum(C[0]) * sum(C[3]) + 4 * sum(C[1]) * sum(C[2]) + 4 * sum(C[2]) * sum(C[3])
  if T == "L":
    R[3] = sum([2 * sum(C[i]) * sum(C[i + 1]) for i in range(n - 1)]) + sum(C[0])**2

  return R

def alternating(v):
  return sum([(-1)**i * d for i, d in enumerate(v)])

def printInfo(T, n):
  print "Cartan matrix for %s_%d" % (T, n)
  C = cartan(T, n)
  print C
  print "dimension of the preprojective algebra"
  print sum(sum(C))

  R = resolution(T, n)
  print "dimension of the projective resolution of the preprojective algebra"
  print R
  print "reality check (= alternating sum of the dimensions)"
  print alternating(R)
  print ""

printInfo("E", 6)
for n in range(2, 6):
  printInfo("A", n)
for n in range(4, 7):
  printInfo("D", n)
for n in range(2, 6):
  printInfo("L", n)