Example #1
0
    def _to_s(self, part):
        r"""
        Returns a function which gives the coefficient of a partition
        in the Schur expansion of ``self(part)``.

        INPUT:

        - ``self`` -- an instance of the Hall-Littlewood `P` basis
        - ``part`` -- a partition

        OUTPUT:

        - returns a function which accepts a second partition ``part2``
          and returns the coefficient of the expansion of the `Qp`
          in the Schur basis.  This is the `t`-Kostka-Foulkes polynomial
          `K_{part2,part}(t)`

        EXAMPLES::

            sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
            sage: HLQp = Sym.hall_littlewood().Qp()
            sage: f21 = HLQp._to_s(Partition([2,1]))
            sage: [f21(p) for p in Partitions(3)]
            [t, 1, 0]
        """
        t = QQt.gen()

        if part == []:
            return lambda part2: QQt.one()

        res = hall_littlewood(part) # call to symmetrica (returns in variable x)
        f = lambda part2: res.coefficient(part2).subs(x=t)
        return f
Example #2
0
    def _to_s(self, part):
        r"""
        Returns a function which gives the coefficient of a partition
        in the Schur expansion of ``self(part)``.

        INPUT:

        - ``self`` -- an instance of the Hall-Littlewood `P` basis
        - ``part`` -- a partition

        OUTPUT:

        - returns a function which accepts a second partition ``part2``
          and returns the coefficient of the expansion of the `Qp`
          in the Schur basis.  This is the `t`-Kostka-Foulkes polynomial
          `K_{part2,part}(t)`

        EXAMPLES::

            sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
            sage: HLQp = Sym.hall_littlewood().Qp()
            sage: f21 = HLQp._to_s(Partition([2,1]))
            sage: [f21(p) for p in Partitions(3)]
            [t, 1, 0]
        """
        t = QQt.gen()

        if part == []:
            return lambda part2: QQt(1)

        res = hall_littlewood(part) # call to symmetrica (returns in variable x)
        f = lambda part2: res.coefficient(part2).subs(x=t)
        return f
Example #3
0
    def _to_s(self, part):
        """
        Returns a function which gives the coefficient of a partition
        in the Schur expansion of ``self(part)``.
        
        EXAMPLES::
        
            sage: Qp = HallLittlewoodQp(QQ)
            sage: f21 = Qp._to_s(Partition([2,1]))
            sage: [f21(p) for p in Partitions(3)]
            [t, 1, 0]
        """
        Zt = ZZ['t']
        t = Zt.gen()
        zero = Zt(0)

        if part == []:
            return lambda part2: Zt(1)

        res = hall_littlewood(part)
        f = lambda part2: res.coefficient(part2).subs(x=t)
        return f
Example #4
0
    def _to_s(self, part):
        """
        Returns a function which gives the coefficient of a partition
        in the Schur expansion of ``self(part)``.
        
        EXAMPLES::
        
            sage: Qp = HallLittlewoodQp(QQ)
            sage: f21 = Qp._to_s(Partition([2,1]))
            sage: [f21(p) for p in Partitions(3)]
            [t, 1, 0]
        """
        Zt = ZZ['t']
        t = Zt.gen()
        zero = Zt(0)

        if part == []:
            return lambda part2: Zt(1)
        
        res = hall_littlewood(part)
        f = lambda part2: res.coefficient(part2).subs(x=t)
        return f