Monomial symmetric polynomials $m_\lambda$

edit · history · discussion · files · short url · bases for symmetric polynomials combinatorics algebra polynomial

Polynomials

$n$

$\lambda$

x1^2

x1^3

x1^4

x1 + x2

x1^2 + x2^2

1, 1:

x1*x2

x1^3 + x2^3

2, 1:

x1^2*x2 + x1*x2^2

x1^4 + x2^4

3, 1:

x1^3*x2 + x1*x2^3

2, 2:

x1^2*x2^2

x1 + x2 + x3

x1^2 + x2^2 + x3^2

1, 1:

x1*x2 + x1*x3 + x2*x3

x1^3 + x2^3 + x3^3

2, 1:

x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2

1, 1, 1:

x1*x2*x3

x1^4 + x2^4 + x3^4

3, 1:

x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3

2, 2:

x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2

2, 1, 1:

x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2

x1 + x2 + x3 + x4

x1^2 + x2^2 + x3^2 + x4^2

1, 1:

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4

x1^3 + x2^3 + x3^3 + x4^3

2, 1:

x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2

1, 1, 1:

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4

x1^4 + x2^4 + x3^4 + x4^4

3, 1:

x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4 + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3

2, 2:

x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2

2, 1, 1:

x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2

1, 1, 1, 1:

x1*x2*x3*x4

x1 + x2 + x3 + x4 + x5

x1^2 + x2^2 + x3^2 + x4^2 + x5^2

1, 1:

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 + x4*x5

x1^3 + x2^3 + x3^3 + x4^3 + x5^3

2, 1:

x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x1^2*x5 + x2^2*x5 + x3^2*x5 + x4^2*x5 + x1*x5^2 + x2*x5^2 + x3*x5^2 + x4*x5^2

1, 1, 1:

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5

x1^4 + x2^4 + x3^4 + x4^4 + x5^4

3, 1:

x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4 + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x1^3*x5 + x2^3*x5 + x3^3*x5 + x4^3*x5 + x1*x5^3 + x2*x5^3 + x3*x5^3 + x4*x5^3

2, 2:

x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2 + x1^2*x5^2 + x2^2*x5^2 + x3^2*x5^2 + x4^2*x5^2

2, 1, 1:

x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 + x1^2*x2*x5 + x1*x2^2*x5 + x1^2*x3*x5 + x2^2*x3*x5 + x1*x3^2*x5 + x2*x3^2*x5 + x1^2*x4*x5 + x2^2*x4*x5 + x3^2*x4*x5 + x1*x4^2*x5 + x2*x4^2*x5 + x3*x4^2*x5 + x1*x2*x5^2 + x1*x3*x5^2 + x2*x3*x5^2 + x1*x4*x5^2 + x2*x4*x5^2 + x3*x4*x5^2

1, 1, 1, 1:

x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5

Definition

For a partition $\lambda$ with at most $n$ parts, the monomial symmetric polynomial is $m_\lambda(x_1, \dots, x_n) = \sum x^{\alpha}$, summed over the distinct rearrangements $\alpha$ of $\lambda$ padded with zeros to length $n$. Each monomial appears once.

Parameters

$n$

— integer ($1 \leq n \leq 5$)

$\lambda$

— Symbolic (Unknown type) (a partition of at most $4$, with at most $n$ parts)

Formulas

(1)

The $m_\lambda$ with $\lambda$ running over partitions of $d$ into at most $n$ parts are a basis of the symmetric polynomials of degree $d$ in $n$ variables.

(2)

$h_d = \sum_{|\lambda| = d} m_\lambda$, where $h_d$ is the complete homogeneous symmetric polynomial Complete_homogeneous_symmetric_polynomials.

(3)

$e_k = m_{(1^k)}$, the partition of $k$ into $k$ ones, where $e_k$ is the elementary symmetric polynomial Elementary_symmetric_polynomials.

(4)

$p_k = m_{(k)}$, the one-part partition, where $p_k$ is the power sum Power_sum_symmetric_polynomials.

(5)

$m_\lambda$ has as many terms as there are distinct rearrangements of $\lambda$ padded to length $n$: $\binom{n}{m_0, m_1, m_2, \dots}$ where $m_i$ counts the parts equal to $i$.

Comments

(6)

The partition is written as its parts, largest first, separated by commas: $2,1,1$ is the partition $(2,1,1)$ of $4$.

(7)

Partitions of at most $4$ in at most $5$ variables, so that the longest entry is 387 characters. The rule across these five tables is the largest complete rectangle whose longest entry stays under about 600.

(8)

The Schur polynomials are indexed by the same partitions and expand in this basis with non-negative integer coefficients, the Kostka numbers.

Programs

(P1)

Sage

from itertools import permutations
def monomial_symmetric(n, lam):
    R = PolynomialRing(ZZ, ['x%d' % (i+1) for i in range(n)])
    exps = list(lam) + [0]*(n - len(lam))
    return sum(prod(g^a for g, a in zip(R.gens(), e))
               for e in sorted(set(permutations(exps))))

monomial_symmetric(5, [3, 2])    # a partition of 5, past this table

Links

[1]

Wikipedia: Symmetric polynomial

Data properties

Entries are of type: integral polynomial

Table is complete: false

How well the digits are known: exact