Schur polynomials $s_\lambda$

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Numbers

$n$

$\lambda$

x1^2

x1^3

x1^4

x1 + x2

x1^2 + x1*x2 + x2^2

1, 1:

x1*x2

x1^3 + x1^2*x2 + x1*x2^2 + x2^3

2, 1:

x1^2*x2 + x1*x2^2

x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4

3, 1:

x1^3*x2 + x1^2*x2^2 + x1*x2^3

2, 2:

x1^2*x2^2

x1 + x2 + x3

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

1, 1:

x1*x2 + x1*x3 + x2*x3

x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3

2, 1:

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

1, 1, 1:

x1*x2*x3

x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4

3, 1:

x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3

2, 2:

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

1, 1:

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

x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3

2, 1:

x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + 2*x1*x2*x4 + x2^2*x4 + 2*x1*x3*x4 + 2*x2*x3*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 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4 + x1^3*x4 + x1^2*x2*x4 + x1*x2^2*x4 + x2^3*x4 + x1^2*x3*x4 + x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x4^4

3, 1:

x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x1^3*x4 + 2*x1^2*x2*x4 + 2*x1*x2^2*x4 + x2^3*x4 + 2*x1^2*x3*x4 + 3*x1*x2*x3*x4 + 2*x2^2*x3*x4 + 2*x1*x3^2*x4 + 2*x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + 2*x1*x2*x4^2 + x2^2*x4^2 + 2*x1*x3*x4^2 + 2*x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3

2, 2:

x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + 2*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*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 + 3*x1*x2*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

Definition

For a partition $\lambda$ with at most $n$ parts, the Schur polynomial is $s_\lambda(x_1, \dots, x_n) = \frac{\det\left(x_i^{\lambda_j + n - j}\right)} {\det\left(x_i^{n-j}\right)}$, the ratio of two determinants, the denominator being the Vandermonde determinant. It is a polynomial despite the division.

Parameters

$n$

— integer ($1 \leq n \leq 4$)

$\lambda$

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

Formulas

(1)

$s_\lambda = \det\left(h_{\lambda_i - i + j}\right)_{1 \leq i,j \leq \ell(\lambda)}$, where $h_d$ are the complete homogeneous symmetric polynomials Complete_homogeneous_symmetric_polynomials and $h_0 = 1$, $h_d = 0$ for $d < 0$ (the Jacobi-Trudi identity). This is how the table is computed.

(2)

$s_{(k)} = h_k$ and $s_{(1^k)} = e_k$, the complete homogeneous and elementary symmetric polynomials Elementary_symmetric_polynomials.

(3)

$s_\lambda = \sum_T x^{T}$, summed over semistandard Young tableaux of shape $\lambda$ with entries in $\{1, \dots, n\}$, where $x^{T}$ records how often each entry occurs.

(4)

$s_\lambda = \sum_\mu K_{\lambda\mu}\, m_\mu$, the monomial symmetric polynomials Monomial_symmetric_polynomials with the Kostka numbers as coefficients.

Comments

(5)

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

(6)

Partitions of at most $4$ in at most $4$ variables, so that the longest entry is 387 characters. These grow faster than the monomial polynomials they expand into, which is why this table stops one variable sooner.

(7)

$s_\lambda$ in $n$ variables is the character of an irreducible polynomial representation of $GL_n$, evaluated at a diagonal matrix with entries $x_1, \dots, x_n$; the partitions with at most $n$ parts index those representations.

Programs

(P1)

Sage

def schur(n, lam):
    R = PolynomialRing(ZZ, ['x%d' % (i+1) for i in range(n)])
    h = lambda d: (R.one() if d == 0 else R.zero() if d < 0 else
                   sum(prod(c) for c in
                       Combinations(list(R.gens())*d, d).list()))
    f = SymmetricFunctions(QQ).s()[lam]
    return R(f.expand(n, alphabet=[str(g) for g in R.gens()]))

schur(4, [3, 2])                 # a partition of 5, past this table

Links

[1]

Wikipedia: Schur polynomial

Data properties

Entries are of type: integral polynomial

Table is complete: false

How well the digits are known: exact