constructing_irreps.sagews

# 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()

[$\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 4$, $\displaystyle 1$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 1 & 0 & -1 \\ 0 & 0 & -1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 3$, $\displaystyle 2$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 0 & -1 & 0 \\ 1 & -1 & 0 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 1$, $\displaystyle 4$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 4$, $\displaystyle 2$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 0 \\ -1 & 0 & 1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 3$, $\displaystyle 1$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 1 \\ 0 & -1 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 2$, $\displaystyle 4$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 0 & 1 & -1 \\ 1 & 0 & -1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 1$, $\displaystyle 3$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 3$, $\displaystyle 4$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 1 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 2$, $\displaystyle 1$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 1 & 0 \\ -1 & 0 & 0 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 4$, $\displaystyle 3$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 1$, $\displaystyle 2$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 2$, $\displaystyle 4$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 3$, $\displaystyle 1$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 1 & -1 & 0 \\ 0 & -1 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 4$, $\displaystyle 2$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 0 & 0 & -1 \\ 1 & 0 & -1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 1$, $\displaystyle 3$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 3$, $\displaystyle 2$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 4$, $\displaystyle 1$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 1 \\ -1 & 0 & 0 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 1$, $\displaystyle 4$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 4$, $\displaystyle 3$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 2$, $\displaystyle 1$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & -1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 3$, $\displaystyle 4$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 0 & 0 \\ -1 & 1 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 4$, $\displaystyle 1$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 1 & 0 & -1 \\ 0 & 0 & -1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 3$, $\displaystyle 2$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 0 & -1 & 0 \\ 1 & -1 & 0 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 1$, $\displaystyle 4$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 4$, $\displaystyle 2$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 0 \\ -1 & 0 & 1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 3$, $\displaystyle 1$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 1 \\ 0 & -1 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 2$, $\displaystyle 4$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 0 & 1 & -1 \\ 1 & 0 & -1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 1$, $\displaystyle 3$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 3$, $\displaystyle 4$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 1 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 2$, $\displaystyle 1$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 1 & 0 \\ -1 & 0 & 0 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 4$, $\displaystyle 3$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 1$, $\displaystyle 2$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 2$, $\displaystyle 4$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 3$, $\displaystyle 1$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 1 \\ 1 & -1 & 0 \\ 0 & -1 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 4$, $\displaystyle 2$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & -1 \\ 0 & 0 & -1 \\ 1 & 0 & -1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 1$, $\displaystyle 3$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 3$, $\displaystyle 2$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 4$, $\displaystyle 1$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 1 & 0 \\ -1 & 0 & 1 \\ -1 & 0 & 0 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 1$, $\displaystyle 4$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 0 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)$

[$\displaystyle 1$, $\displaystyle 4$, $\displaystyle 3$, $\displaystyle 2$] --> $\displaystyle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right)$

[$\displaystyle 3$, $\displaystyle 2$, $\displaystyle 1$, $\displaystyle 4$] --> $\displaystyle \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & -1 \end{array}\right)$

[$\displaystyle 2$, $\displaystyle 3$, $\displaystyle 4$, $\displaystyle 1$] --> $\displaystyle \left(\begin{array}{rrr} 0 & -1 & 0 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{array}\right)$

[$\displaystyle 4$, $\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$] --> $\displaystyle \left(\begin{array}{rrr} -1 & 0 & 1 \\ -1 & 0 & 0 \\ -1 & 1 & 0 \end{array}\right)$