Monomial symmetric functions¶
- class sage.combinat.sf.monomial.SymmetricFunctionAlgebra_monomial(Sym)¶
Bases:
sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical
A class for methods related to monomial symmetric functions
INPUT:
self
– a monomial symmetric function basisSym
– an instance of the ring of the 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: m = SymmetricFunctions(QQ).m() sage: m([2,1]).expand(3) x0^2*x1 + x0*x1^2 + x0^2*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 sage: m([1,1,1]).expand(2) 0 sage: m([2,1]).expand(3,alphabet='z') z0^2*z1 + z0*z1^2 + z0^2*z2 + z1^2*z2 + z0*z2^2 + z1*z2^2 sage: m([2,1]).expand(3,alphabet='x,y,z') x^2*y + x*y^2 + x^2*z + y^2*z + x*z^2 + y*z^2 sage: m([1]).expand(0) 0 sage: (3*m([])).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: m = SymmetricFunctions(QQ).m() sage: (m[3]+m[2,1]+m[1,1,1]).exponential_specialization() 1/6*t^3 sage: x = 5*m[1,1,1] + 3*m[2,1] + 1 sage: x.exponential_specialization() 5/6*t^3 + 1
We also support the \(q\)-exponential_specialization:
sage: factor(m[3].exponential_specialization(q=var("q"), t=var("t"))) (q - 1)^2*t^3/(q^2 + q + 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.
For
q=1
and finiten
we use the formula from Proposition 7.8.3 of [EnumComb2]:\[ps_{n,1}(m_\lambda) = \binom{n}{\ell(\lambda)} \binom{\ell(\lambda)}{m_1(\lambda), m_2(\lambda),\dots},\]where \(\ell(\lambda)\) denotes the length of \(\lambda\).
In all other cases, we convert to complete homogeneous symmetric functions.
EXAMPLES:
sage: m = SymmetricFunctions(QQ).m() sage: x = m[3,1] sage: x.principal_specialization(3) q^7 + q^6 + q^5 + q^3 + q^2 + q sage: x = 5*m[2] + 3*m[1] + 1 sage: x.principal_specialization(3, q=var("q")) -10*(q^3 - 1)*q/(q - 1) + 5*(q^3 - 1)^2/(q - 1)^2 + 3*(q^3 - 1)/(q - 1) + 1
- antipode_by_coercion(element)¶
The antipode of
element
via coercion to and from the power-sum basis or the Schur basis (depending on whether the power sums really form a basis over the given ground ring).INPUT:
element
– element in a basis of the ring of symmetric functions
EXAMPLES:
sage: Sym = SymmetricFunctions(QQ) sage: m = Sym.monomial() sage: m[3,2].antipode() m[3, 2] + 2*m[5] sage: m.antipode_by_coercion(m[3,2]) m[3, 2] + 2*m[5] sage: Sym = SymmetricFunctions(ZZ) sage: m = Sym.monomial() sage: m[3,2].antipode() m[3, 2] + 2*m[5] sage: m.antipode_by_coercion(m[3,2]) m[3, 2] + 2*m[5]
Todo
Is there a not too difficult way to get the power-sum computations to work over any ring, not just one with coercion from \(\QQ\)?
- from_polynomial(f, check=True)¶
Return the symmetric function in the monomial basis corresponding to the polynomial
f
.INPUT:
self
– a monomial symmetric function basisf
– a polynomial in finitely many variables over the same base ring asself
. It is assumed that this polynomial is symmetric.check
– boolean (default:True
), checks whether the polynomial is indeed symmetric
OUTPUT:
This function converts a symmetric polynomial \(f\) in a polynomial ring in finitely many variables to a symmetric function in the monomial basis of the ring of symmetric functions over the same base ring.
EXAMPLES:
sage: m = SymmetricFunctions(QQ).m() sage: P = PolynomialRing(QQ, 'x', 3) sage: x = P.gens() sage: f = x[0] + x[1] + x[2] sage: m.from_polynomial(f) m[1] sage: f = x[0]**2+x[1]**2+x[2]**2 sage: m.from_polynomial(f) m[2] sage: f = x[0]^2+x[1] sage: m.from_polynomial(f) Traceback (most recent call last): ... ValueError: x0^2 + x1 is not a symmetric polynomial sage: f = (m[2,1]+m[1,1]).expand(3) sage: m.from_polynomial(f) m[1, 1] + m[2, 1] sage: f = (2*m[2,1]+m[1,1]+3*m[3]).expand(3) sage: m.from_polynomial(f) m[1, 1] + 2*m[2, 1] + 3*m[3]
- from_polynomial_exp(p)¶
Conversion from polynomial in exponential notation
INPUT:
self
– a monomial symmetric function basisp
– a multivariate polynomial over the same base ring asself
OUTPUT:
This returns a symmetric function by mapping each monomial of \(p\) with exponents
exp
into \(m_\lambda\) where \(\lambda\) is the partition with exponential notationexp
.
EXAMPLES:
sage: m = SymmetricFunctions(QQ).m() sage: P = PolynomialRing(QQ,'x',5) sage: x = P.gens()
The exponential notation of the partition \((5,5,5,3,1,1)\) is:
sage: Partition([5,5,5,3,1,1]).to_exp() [2, 0, 1, 0, 3]
Therefore, the monomial:
sage: f = x[0]^2 * x[2] * x[4]^3
is mapped to:
sage: m.from_polynomial_exp(f) m[5, 5, 5, 3, 1, 1]
Furthermore, this function is linear:
sage: f = 3 * x[3] + 2 * x[0]^2 * x[2] * x[4]^3 sage: m.from_polynomial_exp(f) 3*m[4] + 2*m[5, 5, 5, 3, 1, 1]
See also
- product(left, right)¶
Return the product of
left
andright
.left
,right
– symmetric functions written in the monomial basisself
.
OUTPUT:
the product of
left
andright
, expanded in the monomial basis, as a dictionary whose keys are partitions and whose values are the coefficients of these partitions (more precisely, their respective monomial symmetric functions) in the product.
EXAMPLES:
sage: m = SymmetricFunctions(QQ).m() sage: a = m([2,1]) sage: a^2 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
sage: QQx.<x> = QQ['x'] sage: m = SymmetricFunctions(QQx).m() sage: a = m([2,1])+x sage: 2*a # indirect doctest 2*x*m[] + 2*m[2, 1] sage: a^2 x^2*m[] + 2*x*m[2, 1] + 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]