On the existence of elementwise invariant vectors in representations of symmetric groups

This worksheet provides code to verify the results of the manuscript by Amrutha P, Amritanshu Prasad, and Velmurugan S. Notation here is as in the manuscropt.

It provides the following functions:

w(mu) - this function returns a permutation with cycle type mu, where mu is a given integer partition.
has_invariant(mu,la) - this function returns True if $w_\mu$ admits a non-zero invariant vector in $V_\lambda$ .
is_persistent(la) - this function returns True if $w_\mu$ admits a non-zero invariant vector in $V_\lambda$ for every $\lambda$ except $\lambda=(1^n)$ , where $n$ is the size of $\mu$ .
f(mu) - return the Frobenius characteristic of $\mathrm{Ind}_{C_\mu}^{S_n} 1$ .
mult_triv(n) - return a matrix whose rows and columns are indexed by integer partitions (in reverse lexicographic order. The $(i,j)$ th entry is the multiplicity of $1$ as an eigenvalue of $w_\mu$ in $V_\lambda$ , where $\mu$ and $\lambda$ are the $i$ th and $j$ th partitions, respectively in reverse lexicographic order.

The matrices mult_triv(n) are precompted and stored in the file multiplities.sobj in Sage format and in multiplicities.txt in plain text format for $1\leq n\leq 14$ . This permits verification of the main results for $n\leq 14$ . To carry out this verification, run the file verification.sage using sage. This will print out a list of all pairs of partitions $(\lambda,\mu)$ for which $w_\mu$ does not have a non-zero invariant vector in $V_\lambda$ .

from itertools import accumulate
def w(mu):
    """
    Return a permutation with cycle type `mu`. 
    """
    psmu = list(accumulate(mu))
    nn = list(range(1, sum(mu)+1))
    cycles = [tuple(nn[:psmu[0]])]
    for i,ps in enumerate(psmu[1:]):
        cycles.append(tuple(nn[psmu[i]:ps]))
    return Permutation(cycles)

def has_invariant(mu,la):
    r"""
    Determine whether a permutation with cycle type `\mu` has an invariant vector in `V_\lambda`.
    """
    la = Partition(la)
    Vla = SymmetricGroupRepresentation(la)
    A = Vla.representation_matrix(w(mu))
    return det(A-A.matrix_space().identity_matrix())==0

def is_persistent(mu):
    r"""
    Determine if `\mu` is persistent.
    """
    n = sum(mu)
    return all(has_invariant(mu,la) for la in Partitions(n) if la!=Partition([1]*n))

def f(mu):
    s = SymmetricFunctions(QQ).s()
    return sum(SymmetricGroupRepresentation(la).representation_matrix(w(mu)).eigenvalue_multiplicity(1)*s[la] for la in Partitions(sum(mu)))

def mult_triv(n):
    r"""
    Return the matrix whose `(i,j)`th entry is the multiplicity of the restriction of `V_\lambda` to the cyclic group generated by `w_\mu`, where `\lambda` is the ith partition and `\mu` is the jth partition. 
    """
    T = symmetrica.chartafel(n)
    Pars = list(Partitions(n))
    l = len(Pars)
    A = Matrix(ZZ, l, l)
    for j,mu in enumerate(Partitions(n)):
        wmu = w(mu)
        N = lcm(mu)
        for i,la in enumerate(Partitions(n)):
            A[i,j] = sum(T[i,Pars.index(Partition((wmu^k).cycle_type()))]*euler_phi(N/k) for k in divisors(N))/N
    return A

print(w([2,1])) # print an integer partiton with cycle type (2,1)

[2, 1, 3]

has_invariant([2,1],[1,1,1]) # check if a permutation with cycle type (2,1) has an invariant vector in V_(1,1,1)

False

is_persistent([2,1]) # check if every representation of S_3, with the possible exception of the sign representation, admits a vector invariant under a permutation with cycle type (2,1).

True

f([2,1]) # the Frobenius characteristic of the representation of S_3 induced from the trivial representation of the cyclic group generated by a permutation with cycle type (2,1)

s[2, 1] + s[3]

mult_triv(3) # The (i,j)th entry of this matrix is the multiplicity of 1 as an eigenvalue of a permutation with cycle type mu in V_la where mu and la are the ith and jth partitions of 3 in reverse lexicographic order.

[1 1 1]
[0 1 2]
[1 0 1]