Schur symmetric functions¶

class sage.combinat.sf.schur.SymmetricFunctionAlgebra_schur(Sym)

A class for methods related to the Schur symmetric function basis

INPUT:

• self – a Schur symmetric function basis
• Sym – an instance of the ring of the symmetric functions

TESTS:

sage: s = SymmetricFunctions(QQ).s()
sage: s == loads(dumps(s))
True
sage: TestSuite(s).run(skip=['_test_associativity', '_test_distributivity', '_test_prod'])
sage: TestSuite(s).run(elements = [s[1,1]+s[2], s[1]+2*s[1,1]])

class Element(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
sage: f == loads(dumps(f))
True

expand(n, alphabet='x')

Expand the symmetric function self as a symmetric polynomial in n variables.

INPUT:

• n – a nonnegative integer
• alphabet – (default: 'x') a variable for the expansion

OUTPUT:

A monomial expansion of self in the $$n$$ variables labelled by alphabet.

EXAMPLES:

sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1])
sage: a.expand(2)
x0^2*x1 + x0*x1^2
sage: a.expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2
sage: a.expand(4)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + x0^2*x3 + 2*x0*x1*x3 + x1^2*x3 + 2*x0*x2*x3 + 2*x1*x2*x3 + x2^2*x3 + x0*x3^2 + x1*x3^2 + x2*x3^2
sage: a.expand(2, alphabet='y')
y0^2*y1 + y0*y1^2
sage: a.expand(2, alphabet=['a','b'])
a^2*b + a*b^2
sage: s([1,1,1,1]).expand(3)
0
sage: (s([]) + 2*s([1])).expand(3)
2*x0 + 2*x1 + 2*x2 + 1
sage: s([1]).expand(0)
0
sage: (3*s([])).expand(0)
3

omega()

Return the image of self under the omega automorphism.

The omega automorphism is defined to be the unique algebra endomorphism $$\omega$$ of the ring of symmetric functions that satisfies $$\omega(e_k) = h_k$$ for all positive integers $$k$$ (where $$e_k$$ stands for the $$k$$-th elementary symmetric function, and $$h_k$$ stands for the $$k$$-th complete homogeneous symmetric function). It furthermore is a Hopf algebra endomorphism and an involution, and it is also known as the omega involution. It sends the power-sum symmetric function $$p_k$$ to $$(-1)^{k-1} p_k$$ for every positive integer $$k$$.

The images of some bases under the omega automorphism are given by

$\omega(e_{\lambda}) = h_{\lambda}, \qquad \omega(h_{\lambda}) = e_{\lambda}, \qquad \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad \omega(s_{\lambda}) = s_{\lambda^{\prime}},$

where $$\lambda$$ is any partition, where $$\ell(\lambda)$$ denotes the length (length()) of the partition $$\lambda$$, where $$\lambda^{\prime}$$ denotes the conjugate partition (conjugate()) of $$\lambda$$, and where the usual notations for bases are used ($$e$$ = elementary, $$h$$ = complete homogeneous, $$p$$ = powersum, $$s$$ = Schur).

omega_involution() is a synonym for the omega() method.

OUTPUT:

• the image of self under the omega automorphism

EXAMPLES:

sage: s = SymmetricFunctions(QQ).s()
sage: s([2,1]).omega()
s[2, 1]
sage: s([2,1,1]).omega()
s[3, 1]

omega_involution()

Return the image of self under the omega automorphism.

The omega automorphism is defined to be the unique algebra endomorphism $$\omega$$ of the ring of symmetric functions that satisfies $$\omega(e_k) = h_k$$ for all positive integers $$k$$ (where $$e_k$$ stands for the $$k$$-th elementary symmetric function, and $$h_k$$ stands for the $$k$$-th complete homogeneous symmetric function). It furthermore is a Hopf algebra endomorphism and an involution, and it is also known as the omega involution. It sends the power-sum symmetric function $$p_k$$ to $$(-1)^{k-1} p_k$$ for every positive integer $$k$$.

The images of some bases under the omega automorphism are given by

$\omega(e_{\lambda}) = h_{\lambda}, \qquad \omega(h_{\lambda}) = e_{\lambda}, \qquad \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad \omega(s_{\lambda}) = s_{\lambda^{\prime}},$

where $$\lambda$$ is any partition, where $$\ell(\lambda)$$ denotes the length (length()) of the partition $$\lambda$$, where $$\lambda^{\prime}$$ denotes the conjugate partition (conjugate()) of $$\lambda$$, and where the usual notations for bases are used ($$e$$ = elementary, $$h$$ = complete homogeneous, $$p$$ = powersum, $$s$$ = Schur).

omega_involution() is a synonym for the omega() method.

OUTPUT:

• the image of self under the omega automorphism

EXAMPLES:

sage: s = SymmetricFunctions(QQ).s()
sage: s([2,1]).omega()
s[2, 1]
sage: s([2,1,1]).omega()
s[3, 1]

scalar(x, zee=None)

Return the standard scalar product between self and $$x$$.

Note that the Schur functions are self-dual with respect to this scalar product. They are also lower-triangularly related to the monomial symmetric functions with respect to this scalar product.

INPUT:

• x – element of the ring of symmetric functions over the same base ring as self
• zee – an optional function on partitions giving the value for the scalar product between the power-sum symmetric function $$p_{\mu}$$ and itself (the default value is the standard zee() function)

OUTPUT:

• the scalar product between self and x

EXAMPLES:

sage: s = SymmetricFunctions(ZZ).s()
sage: a = s([2,1])
sage: b = s([1,1,1])
sage: c = 2*s([1,1,1])
sage: d = a + b
sage: a.scalar(a)
1
sage: b.scalar(b)
1
sage: b.scalar(a)
0
sage: b.scalar(c)
2
sage: c.scalar(c)
4
sage: d.scalar(a)
1
sage: d.scalar(b)
1
sage: d.scalar(c)
2

sage: m = SymmetricFunctions(ZZ).monomial()
sage: p4 = Partitions(4)
sage: l = [ [s(p).scalar(m(q)) for q in p4] for p in p4]
sage: matrix(l)
[ 1  0  0  0  0]
[-1  1  0  0  0]
[ 0 -1  1  0  0]
[ 1 -1 -1  1  0]
[-1  2  1 -3  1]

verschiebung(n)

Return the image of the symmetric function self under the $$n$$-th Verschiebung operator.

The $$n$$-th Verschiebung operator $$\mathbf{V}_n$$ is defined to be the unique algebra endomorphism $$V$$ of the ring of symmetric functions that satisfies $$V(h_r) = h_{r/n}$$ for every positive integer $$r$$ divisible by $$n$$, and satisfies $$V(h_r) = 0$$ for every positive integer $$r$$ not divisible by $$n$$. This operator $$\mathbf{V}_n$$ is a Hopf algebra endomorphism. For every nonnegative integer $$r$$ with $$n \mid r$$, it satisfies

$\mathbf{V}_n(h_r) = h_{r/n}, \quad \mathbf{V}_n(p_r) = n p_{r/n}, \quad \mathbf{V}_n(e_r) = (-1)^{r - r/n} e_{r/n}$

(where $$h$$ is the complete homogeneous basis, $$p$$ is the powersum basis, and $$e$$ is the elementary basis). For every nonnegative integer $$r$$ with $$n \nmid r$$, it satisfes

$\mathbf{V}_n(h_r) = \mathbf{V}_n(p_r) = \mathbf{V}_n(e_r) = 0.$

The $$n$$-th Verschiebung operator is also called the $$n$$-th Verschiebung endomorphism. Its name derives from the Verschiebung (German for “shift”) endomorphism of the Witt vectors.

The $$n$$-th Verschiebung operator is adjoint to the $$n$$-th Frobenius operator (see frobenius() for its definition) with respect to the Hall scalar product (scalar()).

The action of the $$n$$-th Verschiebung operator on the Schur basis can also be computed explicitly. The following (probably clumsier than necessary) description can be obtained by solving exercise 7.61 in Stanley’s [STA].

Let $$\lambda$$ be a partition. Let $$n$$ be a positive integer. If the $$n$$-core of $$\lambda$$ is nonempty, then $$\mathbf{V}_n(s_\lambda) = 0$$. Otherwise, the following method computes $$\mathbf{V}_n(s_\lambda)$$: Write the partition $$\lambda$$ in the form $$(\lambda_1, \lambda_2, \ldots, \lambda_{ns})$$ for some nonnegative integer $$s$$. (If $$n$$ does not divide the length of $$\lambda$$, then this is achieved by adding trailing zeroes to $$\lambda$$.) Set $$\beta_i = \lambda_i + ns - i$$ for every $$s \in \{ 1, 2, \ldots, ns \}$$. Then, $$(\beta_1, \beta_2, \ldots, \beta_{ns})$$ is a strictly decreasing sequence of nonnegative integers. Stably sort the list $$(1, 2, \ldots, ns)$$ in order of (weakly) increasing remainder of $$-1 - \beta_i$$ modulo $$n$$. Let $$\xi$$ be the sign of the permutation that is used for this sorting. Let $$\psi$$ be the sign of the permutation that is used to stably sort the list $$(1, 2, \ldots, ns)$$ in order of (weakly) increasing remainder of $$i - 1$$ modulo $$n$$. (Notice that $$\psi = (-1)^{n(n-1)s(s-1)/4}$$.) Then, $$\mathbf{V}_n(s_\lambda) = \xi \psi \prod_{i = 0}^{n - 1} s_{\lambda^{(i)}}$$, where $$(\lambda^{(0)}, \lambda^{(1)}, \ldots, \lambda^{(n - 1)})$$ is the $$n$$-quotient of $$\lambda$$.

INPUT:

• n – a positive integer

OUTPUT:

The result of applying the $$n$$-th Verschiebung operator (on the ring of symmetric functions) to self.

EXAMPLES:

sage: Sym = SymmetricFunctions(ZZ)
sage: s = Sym.s()
sage: s[5].verschiebung(2)
0
sage: s[6].verschiebung(6)
s[1]
sage: s[6,3].verschiebung(3)
s[2, 1] + s[3]
sage: s[6,3,1].verschiebung(2)
-s[3, 2]
sage: s[3,2,1].verschiebung(1)
s[3, 2, 1]
sage: s([]).verschiebung(1)
s[]
sage: s([]).verschiebung(4)
s[]


TESTS:

Let us check that this method on the powersum basis gives the same result as the implementation in sfa.py on the monomial basis:

sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.s(); h = Sym.h()
sage: all( h(s(lam)).verschiebung(3) == h(s(lam).verschiebung(3))
....:      for lam in Partitions(6) )
True
sage: all( s(h(lam)).verschiebung(2) == s(h(lam).verschiebung(2))
....:      for lam in Partitions(4) )
True
sage: all( s(h(lam)).verschiebung(5) == s(h(lam).verschiebung(5))
....:      for lam in Partitions(10) )
True
sage: all( s(h(lam)).verschiebung(2) == s(h(lam).verschiebung(2))
....:      for lam in Partitions(8) )
True
sage: all( s(h(lam)).verschiebung(3) == s(h(lam).verschiebung(3))
....:      for lam in Partitions(12) )
True
sage: all( s(h(lam)).verschiebung(3) == s(h(lam).verschiebung(3))
....:      for lam in Partitions(9) )
True

SymmetricFunctionAlgebra_schur.coproduct_on_basis(mu)

Returns the coproduct of self(mu).

Here self is the basis of Schur functions in the ring of symmetric functions.

INPUT:

• self – a Schur symmetric function basis
• mu – a partition

OUTPUT:

• the image of the mu-th Schur function under the comultiplication of the Hopf algebra of symmetric functions; this is an element of the tensor square of the Schur basis

EXAMPLES:

sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.schur()
sage: s.coproduct_on_basis([2])
s[] # s[2] + s[1] # s[1] + s[2] # s[]


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Power sum symmetric functions

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Symmetric functions, with their multiple realizations