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Identifier
Values
=>
Cc0002;cc-rep
[1]=>1 [2]=>4 [1,1]=>0 [3]=>9 [2,1]=>4 [1,1,1]=>1 [4]=>16 [3,1]=>10 [2,2]=>4 [2,1,1]=>6 [1,1,1,1]=>0 [5]=>25 [4,1]=>18 [3,2]=>11 [3,1,1]=>13 [2,2,1]=>7 [2,1,1,1]=>6 [1,1,1,1,1]=>1 [6]=>36 [5,1]=>28 [4,2]=>20 [4,1,1]=>22 [3,3]=>12 [3,2,1]=>15 [3,1,1,1]=>14 [2,2,2]=>8 [2,2,1,1]=>8 [2,1,1,1,1]=>8 [1,1,1,1,1,1]=>0 [7]=>49 [6,1]=>40 [5,2]=>31 [5,1,1]=>33 [4,3]=>22 [4,2,1]=>25 [4,1,1,1]=>24 [3,3,1]=>17 [3,2,2]=>17 [3,2,1,1]=>17 [3,1,1,1,1]=>17 [2,2,2,1]=>9 [2,2,1,1,1]=>9 [2,1,1,1,1,1]=>8 [1,1,1,1,1,1,1]=>1 [8]=>64 [7,1]=>54 [6,2]=>44 [6,1,1]=>46 [5,3]=>34 [5,2,1]=>37 [5,1,1,1]=>36 [4,4]=>24 [4,3,1]=>28 [4,2,2]=>28 [4,2,1,1]=>28 [4,1,1,1,1]=>28 [3,3,2]=>19 [3,3,1,1]=>19 [3,2,2,1]=>19 [3,2,1,1,1]=>19 [3,1,1,1,1,1]=>18 [2,2,2,2]=>10 [2,2,2,1,1]=>10 [2,2,1,1,1,1]=>10 [2,1,1,1,1,1,1]=>10 [1,1,1,1,1,1,1,1]=>0 [9]=>81 [8,1]=>70 [7,2]=>59 [7,1,1]=>61 [6,3]=>48 [6,2,1]=>51 [6,1,1,1]=>50 [5,4]=>37 [5,3,1]=>41 [5,2,2]=>41
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Description
The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.
References
[1] Reyes, J.-C. U. Random walk, semi-direct products, and card shuffling MathSciNet:2703300
Code
class RandomToRandom:
    def __init__(self, n):
        self._n = n

    @cached_method
    def W(self):
        return SymmetricGroup(self._n)

    def cycle(self, i,j):
        """
        EXAMPLES::

            sage: RandomToRandom(5).cycle(2,4)
            (2,3,4)
        """
        return self.W()([tuple(range(i,j+1))])

    def r2r(self, i,j):
        """
        EXAMPLES::

            sage: R = RandomToRandom(4)
            sage: R.r2r(1,3,4)
            (1,2,4)
            sage: R.r2r(3,1,4)
            (1,4,2)
            sage: r2r(1,2,4)
            (1,4)

            sage: RandomToRandom(3).r2r(1,2,3)
            (1,3)
        """
        return self.cycle(i,self._n) * (~self.cycle(j,self._n))

    def operator(self, representation):
        """
        EXAMPLES::

            sage: R = RandomToRandom(3)
            sage: representation = attrcall("matrix") # emulates the natural representation
            sage: R.operator(representation)
            [5 1 3]
            [1 5 3]
            [3 3 3]

            sage: representation = R.W().algebra(QQ).monomial         # emulates the regular representation

            sage: R.operator(representation)
            3*B[()] + 2*B[(2,3)] + B[(1,2,3)] + B[(1,3,2)] + 2*B[(1,3)]

            sage: R.operator(Partition([2,1]))
            [2 2]
            [2 2]
        """
        if isinstance(representation, Partition):
            assert representation.size() == self._n
            representation = SymmetricGroupRepresentation(representation)
        E = self.W().domain()
        return sum(representation(self.r2r(i,j)) for i in E for j in E)

    def max_eigenvalue_on_simple_representation(self, p):
        return max(self.operator(p).eigenvalues())

    def max_eigenvalue_on_simple_representations(self):
        """
        EXAMPLES::

            sage: R = RandomToRandom(4)
            sage: dict(R.max_eigenvalue_on_simple_representations())
            {[1, 1, 1, 1]: 0, [2, 1, 1]: 6, [2, 2]: 4, [3, 1]: 10, [4]: 16}

            sage: R = RandomToRandom(5)
            sage: dict(R.max_eigenvalue_on_simple_representations())
            {[1, 1, 1, 1]: 0, [2, 1, 1]: 6, [2, 2]: 4, [3, 1]: 10, [4]: 16}
        """
        from sage.sets.family import Family
        return Family(Partitions(self._n), self.max_eigenvalue_on_simple_representation)

@cached_function
def getRandomToRandom(n):
    return RandomToRandom(n)

def statistic(L):
    n = sum(L)
    R = getRandomToRandom(n)
    d = R.max_eigenvalue_on_simple_representations()
    return d[L]
Created
Mar 22, 2013 at 22:05 by Nicolas M. Thiéry
Updated
Dec 29, 2016 at 09:25 by Christian Stump