Homogeneous symmetric functions¶
By this we mean the basis formed of the complete homogeneous symmetric functions \(h_\lambda\), not an arbitrary graded basis.
- class sage.combinat.sf.homogeneous.SymmetricFunctionAlgebra_homogeneous(Sym)¶
Bases:
sage.combinat.sf.multiplicative.SymmetricFunctionAlgebra_multiplicative
A class of methods specific to the homogeneous basis of symmetric functions.
INPUT:
self
– a homogeneous basis of symmetric functionsSym
– an instance of the ring of symmetric functions
- class Element¶
Bases:
sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical.Element
- expand(n, alphabet='x')¶
Expand the symmetric function
self
as a symmetric polynomial inn
variables.INPUT:
n
– a nonnegative integeralphabet
– (default:'x'
) a variable for the expansion
OUTPUT:
A monomial expansion of
self
in the \(n\) variables labelled byalphabet
.EXAMPLES:
sage: h = SymmetricFunctions(QQ).h() sage: h([3]).expand(2) x0^3 + x0^2*x1 + x0*x1^2 + x1^3 sage: h([1,1,1]).expand(2) x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3 sage: h([2,1]).expand(3) x0^3 + 2*x0^2*x1 + 2*x0*x1^2 + x1^3 + 2*x0^2*x2 + 3*x0*x1*x2 + 2*x1^2*x2 + 2*x0*x2^2 + 2*x1*x2^2 + x2^3 sage: h([3]).expand(2,alphabet='y') y0^3 + y0^2*y1 + y0*y1^2 + y1^3 sage: h([3]).expand(2,alphabet='x,y') x^3 + x^2*y + x*y^2 + y^3 sage: h([3]).expand(3,alphabet='x,y,z') x^3 + x^2*y + x*y^2 + y^3 + x^2*z + x*y*z + y^2*z + x*z^2 + y*z^2 + z^3 sage: (h([]) + 2*h([1])).expand(3) 2*x0 + 2*x1 + 2*x2 + 1 sage: h([1]).expand(0) 0 sage: (3*h([])).expand(0) 3
- exponential_specialization(t=None, q=1)¶
Return the exponential specialization of a symmetric function (when \(q = 1\)), or the \(q\)-exponential specialization (when \(q \neq 1\)).
The exponential specialization \(ex\) at \(t\) is a \(K\)-algebra homomorphism from the \(K\)-algebra of symmetric functions to another \(K\)-algebra \(R\). It is defined whenever the base ring \(K\) is a \(\QQ\)-algebra and \(t\) is an element of \(R\). The easiest way to define it is by specifying its values on the powersum symmetric functions to be \(p_1 = t\) and \(p_n = 0\) for \(n > 1\). Equivalently, on the homogeneous functions it is given by \(ex(h_n) = t^n / n!\); see Proposition 7.8.4 of [EnumComb2].
By analogy, the \(q\)-exponential specialization is a \(K\)-algebra homomorphism from the \(K\)-algebra of symmetric functions to another \(K\)-algebra \(R\) that depends on two elements \(t\) and \(q\) of \(R\) for which the elements \(1 - q^i\) for all positive integers \(i\) are invertible. It can be defined by specifying its values on the complete homogeneous symmetric functions to be
\[ex_q(h_n) = t^n / [n]_q!,\]where \([n]_q!\) is the \(q\)-factorial. Equivalently, for \(q \neq 1\) and a homogeneous symmetric function \(f\) of degree \(n\), we have
\[ex_q(f) = (1-q)^n t^n ps_q(f),\]where \(ps_q(f)\) is the stable principal specialization of \(f\) (see
principal_specialization()
). (See (7.29) in [EnumComb2].)The limit of \(ex_q\) as \(q \to 1\) is \(ex\).
INPUT:
t
(default:None
) – the value to use for \(t\); the default is to create a ring of polynomials int
.q
(default: \(1\)) – the value to use for \(q\). Ifq
isNone
, then a ring (or fraction field) of polynomials inq
is created.
EXAMPLES:
sage: h = SymmetricFunctions(QQ).h() sage: x = h[5,3] sage: x.exponential_specialization() 1/720*t^8 sage: factorial(5)*factorial(3) 720 sage: x = 5*h[1,1,1] + 3*h[2,1] + 1 sage: x.exponential_specialization() 13/2*t^3 + 1
We also support the \(q\)-exponential_specialization:
sage: factor(h[3].exponential_specialization(q=var("q"), t=var("t"))) t^3/((q^2 + q + 1)*(q + 1))
- 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 theomega()
method.OUTPUT:
the image of
self
under the omega automorphism
EXAMPLES:
sage: h = SymmetricFunctions(QQ).h() sage: a = h([2,1]); a h[2, 1] sage: a.omega() h[1, 1, 1] - h[2, 1] sage: e = SymmetricFunctions(QQ).e() sage: e(h([2,1]).omega()) e[2, 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 theomega()
method.OUTPUT:
the image of
self
under the omega automorphism
EXAMPLES:
sage: h = SymmetricFunctions(QQ).h() sage: a = h([2,1]); a h[2, 1] sage: a.omega() h[1, 1, 1] - h[2, 1] sage: e = SymmetricFunctions(QQ).e() sage: e(h([2,1]).omega()) e[2, 1]
- principal_specialization(n=+ Infinity, q=None)¶
Return the principal specialization of a symmetric function.
The principal specialization of order \(n\) at \(q\) is the ring homomorphism \(ps_{n,q}\) from the ring of symmetric functions to another commutative ring \(R\) given by \(x_i \mapsto q^{i-1}\) for \(i \in \{1,\dots,n\}\) and \(x_i \mapsto 0\) for \(i > n\). Here, \(q\) is a given element of \(R\), and we assume that the variables of our symmetric functions are \(x_1, x_2, x_3, \ldots\). (To be more precise, \(ps_{n,q}\) is a \(K\)-algebra homomorphism, where \(K\) is the base ring.) See Section 7.8 of [EnumComb2].
The stable principal specialization at \(q\) is the ring homomorphism \(ps_q\) from the ring of symmetric functions to another commutative ring \(R\) given by \(x_i \mapsto q^{i-1}\) for all \(i\). This is well-defined only if the resulting infinite sums converge; thus, in particular, setting \(q = 1\) in the stable principal specialization is an invalid operation.
INPUT:
n
(default:infinity
) – a nonnegative integer orinfinity
, specifying whether to compute the principal specialization of ordern
or the stable principal specialization.q
(default:None
) – the value to use for \(q\); the default is to create a ring of polynomials inq
(or a field of rational functions inq
) over the given coefficient ring.
We use the formulas from Proposition 7.8.3 of [EnumComb2] (using Gaussian binomial coefficients \(\binom{u}{v}_q\)):
\[ \begin{align}\begin{aligned}ps_{n,q}(h_\lambda) = \prod_i \binom{n+\lambda_i-1}{\lambda_i}_q,\\ps_{n,1}(h_\lambda) = \prod_i \binom{n+\lambda_i-1}{\lambda_i},\\ps_q(h_\lambda) = 1 / \prod_i \prod_{j=1}^{\lambda_i} (1-q^j).\end{aligned}\end{align} \]EXAMPLES:
sage: h = SymmetricFunctions(QQ).h() sage: x = h[2,1] sage: x.principal_specialization(3) q^6 + 2*q^5 + 4*q^4 + 4*q^3 + 4*q^2 + 2*q + 1 sage: x = 3*h[2] + 2*h[1] + 1 sage: x.principal_specialization(3, q=var("q")) 2*(q^3 - 1)/(q - 1) + 3*(q^4 - 1)*(q^3 - 1)/((q^2 - 1)*(q - 1)) + 1
- coproduct_on_generators(i)¶
Return the coproduct on \(h_i\).
INPUT:
self
– a homogeneous basis of symmetric functionsi
– a nonnegative integer
OUTPUT:
the sum \(\sum_{r=0}^i h_r \otimes h_{i-r}\)
EXAMPLES:
sage: Sym = SymmetricFunctions(QQ) sage: h = Sym.homogeneous() sage: h.coproduct_on_generators(2) h[] # h[2] + h[1] # h[1] + h[2] # h[] sage: h.coproduct_on_generators(0) h[] # h[]