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Author: David Renshaw
Views : 85
License: MIT License
Description: notes on Young Symmetrizers
Compute Environment: Ubuntu 20.04 (Default)
# Does `g` map each element in `lst` to another element in `lst`? def stabilizes(g, lst): for x in lst: if lst.count(g(x)) == 0: return False return True # Does `g` stabilize all parts of the standard tableau `t`? def stabilizes_all(g, t): for lst in t: if not stabilizes(g, lst): return False return True # See https://en.wikipedia.org/wiki/Young_symmetrizer # `t` is a standard tableau def young_symmetrizer(t): G = SymmetricGroup(t.size()) A = GroupAlgebra(G, QQ) a = A(0) b = A(0) for g in G: if stabilizes_all(g, t): a = a + A(g) s = t.conjugate() if stabilizes_all(g, s): b = b + g.sign() * A(g) return a * b # Compute matrices for the representation given by a standard tableau `t`. # Returns a dictionary, mapping group elements to matrices. def compute_representation(t): G = SymmetricGroup(t.size()) c = young_symmetrizer(t) basis = [] # basis vectors basis_in_algebra = [] # basis vectors as elements of the group algebra for g in G: z = g * c zv = z.to_vector() if Matrix(basis + [zv]).rank() > len(basis): # not in the span yet basis.append(zv) basis_in_algebra.append(z) basis_matrix = Matrix(basis) g_to_m = {} for g in G: ms = [] # see what g does on each basis vector, and map it back to the basis's coordinates for ba in basis_in_algebra: ms.append(basis_matrix.transpose() \ (g * ba).to_vector()) msm = Matrix(ms).transpose() g_to_m[g] = msm return g_to_m
for t in StandardTableaux(4): print("Tableau = {}".format(t)) g_to_m = compute_representation(t) for (g,m) in g_to_m.items(): for (g1,m1) in g_to_m.items(): assert(g_to_m[g * g1] == m * m1) print("OK")
Tableau = [[1, 2, 3, 4]] Tableau = [[1, 3, 4], [2]] Tableau = [[1, 2, 4], [3]] Tableau = [[1, 2, 3], [4]] Tableau = [[1, 3], [2, 4]] Tableau = [[1, 2], [3, 4]] Tableau = [[1, 4], [2], [3]] Tableau = [[1, 3], [2], [4]] Tableau = [[1, 2], [3], [4]] Tableau = [[1], [2], [3], [4]] OK
sage.misc.latex.EMBEDDED_MODE = True g_to_m = compute_representation(StandardTableaux(4)([[1, 2, 3], [4]])) to_display = [] for (g,m) in g_to_m.items(): # take inverse of g because Sage permutations act on the right view(g.inverse()([1,2,3,4]), "-->", m) print()
[1\displaystyle 1, 2\displaystyle 2, 3\displaystyle 3, 4\displaystyle 4] --> (100010001)\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)
[3\displaystyle 3, 4\displaystyle 4, 1\displaystyle 1, 2\displaystyle 2] --> (011101001)\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 1 & 0 & -1 \\ 0 & 0 & -1 \end{array}\right)
[4\displaystyle 4, 3\displaystyle 3, 2\displaystyle 2, 1\displaystyle 1] --> (011010110)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 0 & -1 & 0 \\ 1 & -1 & 0 \end{array}\right)
[2\displaystyle 2, 1\displaystyle 1, 4\displaystyle 4, 3\displaystyle 3] --> (100101110)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right)
[1\displaystyle 1, 4\displaystyle 4, 2\displaystyle 2, 3\displaystyle 3] --> (110100101)\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 0 \\ -1 & 0 & 1 \end{array}\right)
[2\displaystyle 2, 3\displaystyle 3, 1\displaystyle 1, 4\displaystyle 4] --> (110011010)\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 1 \\ 0 & -1 & 0 \end{array}\right)
[3\displaystyle 3, 2\displaystyle 2, 4\displaystyle 4, 1\displaystyle 1] --> (001011101)\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 0 & 1 & -1 \\ 1 & 0 & -1 \end{array}\right)
[4\displaystyle 4, 1\displaystyle 1, 3\displaystyle 3, 2\displaystyle 2] --> (001100010)\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)
[1\displaystyle 1, 3\displaystyle 3, 4\displaystyle 4, 2\displaystyle 2] --> (010110011)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 1 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)
[4\displaystyle 4, 2\displaystyle 2, 1\displaystyle 1, 3\displaystyle 3] --> (101110100)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 1 & 0 \\ -1 & 0 & 0 \end{array}\right)
[2\displaystyle 2, 4\displaystyle 4, 3\displaystyle 3, 1\displaystyle 1] --> (010001100)\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right)
[3\displaystyle 3, 1\displaystyle 1, 2\displaystyle 2, 4\displaystyle 4] --> (101001011)\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)
[1\displaystyle 1, 2\displaystyle 2, 4\displaystyle 4, 3\displaystyle 3] --> (100110101)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right)
[4\displaystyle 4, 3\displaystyle 3, 1\displaystyle 1, 2\displaystyle 2] --> (011110010)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 1 & -1 & 0 \\ 0 & -1 & 0 \end{array}\right)
[3\displaystyle 3, 4\displaystyle 4, 2\displaystyle 2, 1\displaystyle 1] --> (011001101)\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 0 & 0 & -1 \\ 1 & 0 & -1 \end{array}\right)
[2\displaystyle 2, 1\displaystyle 1, 3\displaystyle 3, 4\displaystyle 4] --> (100001010)\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right)
[1\displaystyle 1, 3\displaystyle 3, 2\displaystyle 2, 4\displaystyle 4] --> (110010011)\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)
[2\displaystyle 2, 4\displaystyle 4, 1\displaystyle 1, 3\displaystyle 3] --> (110101100)\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 1 \\ -1 & 0 & 0 \end{array}\right)
[4\displaystyle 4, 2\displaystyle 2, 3\displaystyle 3, 1\displaystyle 1] --> (001010100)\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right)
[3\displaystyle 3, 1\displaystyle 1, 4\displaystyle 4, 2\displaystyle 2] --> (001101011)\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)
[1\displaystyle 1, 4\displaystyle 4, 3\displaystyle 3, 2\displaystyle 2] --> (010100001)\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right)
[3\displaystyle 3, 2\displaystyle 2, 1\displaystyle 1, 4\displaystyle 4] --> (101011001)\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & -1 \end{array}\right)
[2\displaystyle 2, 3\displaystyle 3, 4\displaystyle 4, 1\displaystyle 1] --> (010011110)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{array}\right)
[4\displaystyle 4, 1\displaystyle 1, 2\displaystyle 2, 3\displaystyle 3] --> (101100110)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 0 & 0 \\ -1 & 1 & 0 \end{array}\right)
[3\displaystyle 3, 4\displaystyle 4, 1\displaystyle 1, 2\displaystyle 2] --> (011101001)\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 1 & 0 & -1 \\ 0 & 0 & -1 \end{array}\right)
[4\displaystyle 4, 3\displaystyle 3, 2\displaystyle 2, 1\displaystyle 1] --> (011010110)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 0 & -1 & 0 \\ 1 & -1 & 0 \end{array}\right)
[2\displaystyle 2, 1\displaystyle 1, 4\displaystyle 4, 3\displaystyle 3] --> (100101110)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right)
[1\displaystyle 1, 4\displaystyle 4, 2\displaystyle 2, 3\displaystyle 3] --> (110100101)\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 0 \\ -1 & 0 & 1 \end{array}\right)
[2\displaystyle 2, 3\displaystyle 3, 1\displaystyle 1, 4\displaystyle 4] --> (110011010)\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 1 \\ 0 & -1 & 0 \end{array}\right)
[3\displaystyle 3, 2\displaystyle 2, 4\displaystyle 4, 1\displaystyle 1] --> (001011101)\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 0 & 1 & -1 \\ 1 & 0 & -1 \end{array}\right)
[4\displaystyle 4, 1\displaystyle 1, 3\displaystyle 3, 2\displaystyle 2] --> (001100010)\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)
[1\displaystyle 1, 3\displaystyle 3, 4\displaystyle 4, 2\displaystyle 2] --> (010110011)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 1 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)
[4\displaystyle 4, 2\displaystyle 2, 1\displaystyle 1, 3\displaystyle 3] --> (101110100)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 1 & 0 \\ -1 & 0 & 0 \end{array}\right)
[2\displaystyle 2, 4\displaystyle 4, 3\displaystyle 3, 1\displaystyle 1] --> (010001100)\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right)
[3\displaystyle 3, 1\displaystyle 1, 2\displaystyle 2, 4\displaystyle 4] --> (101001011)\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)
[1\displaystyle 1, 2\displaystyle 2, 4\displaystyle 4, 3\displaystyle 3] --> (100110101)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right)
[4\displaystyle 4, 3\displaystyle 3, 1\displaystyle 1, 2\displaystyle 2] --> (011110010)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 1 & -1 & 0 \\ 0 & -1 & 0 \end{array}\right)
[3\displaystyle 3, 4\displaystyle 4, 2\displaystyle 2, 1\displaystyle 1] --> (011001101)\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 0 & 0 & -1 \\ 1 & 0 & -1 \end{array}\right)
[2\displaystyle 2, 1\displaystyle 1, 3\displaystyle 3, 4\displaystyle 4] --> (100001010)\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right)
[1\displaystyle 1, 3\displaystyle 3, 2\displaystyle 2, 4\displaystyle 4] --> (110010011)\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)
[2\displaystyle 2, 4\displaystyle 4, 1\displaystyle 1, 3\displaystyle 3] --> (110101100)\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 1 \\ -1 & 0 & 0 \end{array}\right)
[4\displaystyle 4, 2\displaystyle 2, 3\displaystyle 3, 1\displaystyle 1] --> (001010100)\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right)
[3\displaystyle 3, 1\displaystyle 1, 4\displaystyle 4, 2\displaystyle 2] --> (001101011)\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)
[1\displaystyle 1, 4\displaystyle 4, 3\displaystyle 3, 2\displaystyle 2] --> (010100001)\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right)
[3\displaystyle 3, 2\displaystyle 2, 1\displaystyle 1, 4\displaystyle 4] --> (101011001)\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & -1 \end{array}\right)
[2\displaystyle 2, 3\displaystyle 3, 4\displaystyle 4, 1\displaystyle 1] --> (010011110)\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{array}\right)
[4\displaystyle 4, 1\displaystyle 1, 2\displaystyle 2, 3\displaystyle 3] --> (101100110)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 0 & 0 \\ -1 & 1 & 0 \end{array}\right)