|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''Schur polynomials''', named after [[Issai Schur]], are certain [[symmetric polynomial]]s in ''n'' variables, indexed by [[integer partition|partition]]s, that generalize the [[elementary symmetric polynomial]]s and the [[complete homogeneous symmetric polynomial]]s. In [[representation theory]] they are the characters of [[irreducible representation]]s of the [[general linear group]]s. The Schur polynomials form a [[basis (linear algebra)|linear basis]] for the space of all symmetric polynomials. Any product of Schur functions can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the [[Littlewood-Richardson rule]]. More generally, '''skew Schur polynomials''' are associated with pairs of partitions and have similar properties to Schur polynomials.
| | The writer is known as Wilber Pegues. My working day occupation is an information officer but I've already applied for an additional 1. The favorite hobby for him and his children is to perform lacross and he'll be starting something else alongside with it. For many years he's been living in Mississippi and he doesn't plan on changing it.<br><br>Also visit my website :: [http://kjhkkb.net/xe/notice/374835 real psychics] |
| | |
| ==Definition==
| |
| | |
| Schur polynomials correspond to [[integer partition]]s. Given a partition
| |
| | |
| :<math> d = d_1 + d_2 + \cdots + d_n, \; \; d_1 \geq d_2 \geq \cdots \ge d_n</math>
| |
| | |
| (where each ''d''<sub>''j''</sub> is a non-negative integer), the following functions are [[alternating polynomials]] (in other words they change sign under any [[transposition (mathematics)|transposition]] of the variables):
| |
| | |
| :<math> a_{(d_1+n-1, d_2+n-2, \dots , d_n)} (x_1, x_2, \dots , x_n) =
| |
| \det \left[ \begin{matrix} x_1^{d_1+n-1} & x_2^{d_1+n-1} & \dots & x_n^{d_1+n-1} \\
| |
| x_1^{d_2+n-2} & x_2^{d_2+n-2} & \dots & x_n^{d_2+n-2} \\
| |
| \vdots & \vdots & \ddots & \vdots \\
| |
| x_1^{d_n} & x_2^{d_n} & \dots & x_n^{d_n} \end{matrix} \right]
| |
| </math>
| |
| | |
| Since they are alternating, they are all divisible by the [[Vandermonde determinant]]:
| |
| | |
| :<math> a_{(n-1, n-2, \dots , 0)} (x_1, x_2, \dots , x_n) = \det \left[ \begin{matrix} x_1^{n-1} & x_2^{n-1} & \dots & x_n^{n-1} \\
| |
| x_1^{n-2} & x_2^{n-2} & \dots & x_n^{n-2} \\
| |
| \vdots & \vdots & \ddots & \vdots \\
| |
| 1 & 1 & \dots & 1 \end{matrix} \right] = \prod_{1 \leq j < k \leq n} (x_j-x_k). </math>
| |
| The Schur polynomials are defined as the ratio:
| |
| :<math>
| |
| s_{(d_1, d_2, \dots , d_n)} (x_1, x_2, \dots , x_n) =
| |
| \frac{ a_{(d_1+n-1, d_2+n-2, \dots , d_n+0)} (x_1, x_2, \dots , x_n)}
| |
| {a_{(n-1, n-2, \dots , 0)} (x_1, x_2, \dots , x_n) }. </math>
| |
| | |
| This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the Vandermonde determinant.
| |
| | |
| ==Properties==
| |
| The degree ''d'' Schur polynomials in ''n'' variables are a linear basis for the space of homogeneous degree ''d'' symmetric polynomials in ''n'' variables.
| |
| | |
| For a partition λ, the Schur polynomial is a sum of monomials:
| |
| | |
| :<math>
| |
| S_\lambda(x_1,x_2,\ldots,x_n)=\sum_T x^T = \sum_T x_1^{t_1}\cdots x_n^{t_n}
| |
| </math>
| |
| | |
| where the summation is over all semistandard [[Young tableau]]x ''T'' of shape λ; the exponents ''t''<sub>1</sub>, ..., ''t''<sub>''n''</sub> give the weight of ''T'', in other words each ''t''<sub>''i''</sub> counts the occurrences of the number ''i'' in ''T''. This can be shown to be equivalent to the definition from the first Giambelli formula using the [[Lindström–Gessel–Viennot lemma]] (as outlined on that page).
| |
| | |
| The first [[Jacobi]]-Trudi formula expresses the Schur polynomial as a determinant
| |
| in terms of the [[complete homogeneous symmetric polynomial]]s:
| |
| | |
| :<math> S_{\lambda} = \det_{ij} h_{\lambda_{i} + j - i}. </math>
| |
| where
| |
| :<math> h_i := S_{(i)} </math>.
| |
|
| |
| The second Jacobi-Trudi formula expresses the Schur polynomial as | |
| a determinant in terms of the [[elementary symmetric polynomial]]s:
| |
| | |
| :<math> S_{\lambda} = \det_{ij} e_{\lambda'_{i} + j - i} </math>,
| |
| where
| |
| :<math> e_i := S_{(1)^j} </math>
| |
| | |
| where <math>\lambda'</math> is the dual partition to <math>\lambda</math>.
| |
| | |
| These two formulae are known as “determinantal identities". Another such identity is
| |
| the [[Giambelli]] formula, which expresses the Schur function for an arbitrary partition
| |
| in terms of those for the “hook partitions“ contained within the Young diagram.
| |
| In Frobenius notation, the partition is denoted
| |
| :<math> (a_{1}, ... a_{r}| b_{1}, ... b_{r})</math>
| |
| | |
| where, for each diagonal element in position <math> ii </math>, <math> a_{i} </math> denotes
| |
| the number of boxes to the right in the same row and <math> b_{i} </math> denotes the number of boxes beneath it
| |
| in the same column (the “arm“ and “leg“ lengths, respectively).
| |
| | |
| The [[Giambelli]] identity expresses the partition as the determinant
| |
| :<math> S_{ (a_{1}, ... a_{r}| b_{1}, ... b_{r})} = \det ( S_{(a_{i} | b_{j})}) </math>.
| |
| | |
| Schur polynomials can be expressed as linear combinations of [[Symmetric polynomial#Monomial symmetric polynomials|monomial symmetric functions]] ''m''<sub>μ</sub> with non-negative integer coefficients ''K''<sub>λμ</sub> called [[Kostka number]]s:
| |
| | |
| : <math>S_\lambda= \sum_\mu K_{\lambda\mu}m_\mu.\ </math>
| |
| | |
| Evaluating the Schur polynomial ''S''<sub>λ</sub> in (1,1,...,1) gives the number of semi-standard Young tableaux of shape λ with entries in 1,2,...n.
| |
| One can show, by using the Weyl character formula for example, that
| |
| | |
| : <math>S_\lambda(1,1,\dots,1) = \prod_{1\leq i < j \leq n} \frac{\lambda_i - \lambda_j + j-i}{j-i}.</math>
| |
| | |
| In this formula, λ, the tuple indicating the width of each row of the Young diagram, is implicitly extended with zeros until it has length <math>n</math>. The sum of the elements <math>\lambda_i</math> is <math>d</math>.
| |
| | |
| ==Example==
| |
| The following extended example should help clarify these ideas. Consider the case ''n'' = 3, ''d'' = 4. Using Ferrers diagrams or some other method, we find that there are just four partitions of 4 into at most three parts. We have
| |
| | |
| :<math> S_{(2,1,1)} (x_1, x_2, x_3) = \frac{1}{\Delta} \;
| |
| \det \left[ \begin{matrix} x_1^4 & x_2^4 & x_3^4 \\ x_1^2 & x_2^2 & x_3^2 \\ x_1 & x_2 & x_3 \end{matrix}
| |
| \right] = x_1 \, x_2 \, x_3 \, (x_1 + x_2 + x_3) </math>
| |
| | |
| :<math> S_{(2,2,0)} (x_1, x_2, x_3) = \frac{1}{\Delta} \;
| |
| \det \left[ \begin{matrix} x_1^4 & x_2^4 & x_3^4 \\ x_1^3 & x_2^3 & x_3^3 \\ 1 & 1 & 1 \end{matrix}
| |
| \right]= x_1^2 \, x_2^2 + x_1^2 \, x_3^2 + x_2^2 \, x_3^2
| |
| + x_1^2 \, x_2 \, x_3 + x_1 \, x_2^2 \, x_3 + x_1 \, x_2 \, x_3^2 </math>
| |
| | |
| and so on. Summarizing:
| |
| | |
| #<math> S_{(2,1,1)} = e_1 \, e_3</math>
| |
| #<math> S_{(2,2,0)} = e_2^2 - e_1 \, e_3</math>
| |
| #<math> S_{(3,1,0)} = e_1^2 \, e_2 - e_2^2 - e_1 \, e_3</math>
| |
| #<math> S_{(4,0,0)} = e_1^4 - 3 \, e_1^2 \, e_2 + 2 \, e_1 \, e_3 + e_2^2.</math>
| |
| | |
| Every homogeneous degree-four symmetric polynomial in three variables can be expressed as a unique ''linear combination'' of these four Schur polynomials, and this combination can again be found using a Gröbner basis for an appropriate elimination order. For example,
| |
| | |
| :<math>\phi(x_1, x_2, x_3) = x_1^4 + x_2^4 + x_3^4</math>
| |
| | |
| is obviously a symmetric polynomial which is homogeneous of degree four, and we have
| |
| | |
| :<math>\phi = S_{(2,1,1)} - S_{(3,1,0)} + S_{(4,0,0)}.\,\!</math>
| |
| | |
| ==Relation to representation theory==
| |
| | |
| The Schur polynomials occur in the [[representation theory of the symmetric group]]s, [[general linear group]]s, and [[unitary group]]s. The [[Weyl character formula]] implies that the Schur polynomials are the characters of finite dimensional irreducible representations of the general linear groups, and helps to generalize Schur's work to other compact and semisimple [[Lie group]]s.
| |
| | |
| Several expressions arise for this relation, one of the most important being the expansion of the Schur functions ''s''<sub>λ</sub> in terms of the symmetric power functions <math>p_k=\sum_i x_i^k</math>. If we write χ{{su|p=λ|b=ρ}} for the character of the representation of the symmetric group indexed by the partition λ evaluated at elements of cycle type indexed by the partition ρ, then
| |
| :<math>s_\lambda=\sum_{\rho=(1^{r_1},2^{r_2},3^{r_3},\dots)}\chi^\lambda_\rho \prod_k \frac{p^{r_k}_k}{r_k!},</math>
| |
| where ρ = (1<sup>''r''<sub>1</sub></sup>, 2<sup>''r''<sub>2</sub></sup>, 3<sup>''r''<sub>3</sub></sup>, ...) means that the partition ρ has ''r''<sub>''k''</sub> parts of length ''k''.
| |
| | |
| ==Skew Schur functions==
| |
| Skew Schur functions ''s''<sub>λ/μ</sub> depend on two partitions λ and μ, and can be defined by the property
| |
| :<math>\langle s_{\lambda/\mu},s_\nu\rangle = \langle s_{\lambda},s_\mu s_\nu\rangle. </math> | |
| | |
| | |
| Similar to the ordinary Schur polynomials, there are numerous ways to compute these. The corresponding Jacobi-Trudi identities are
| |
| :<math>S_{\lambda/\mu} = (h_{\lambda_i - \mu_j -i + j}), 1\leq i,j \leq l(\lambda)</math>,
| |
| :<math>S_{\lambda'/\mu'} = (e_{\lambda_i - \mu_j -i + j}), 1\leq i,j \leq l(\lambda)</math>. | |
| | |
| There is also a combinatorial interpretation of the skew Schur polynomials,
| |
| namely it is a sum over all semi-standard Young tableaux (or column-strict tableaux) of the skew shape <math>\lambda/\mu</math>.
| |
| | |
| ==See also==
| |
| | |
| *[[Littlewood-Richardson rule]], where one finds some identities involving Schur polynomials.
| |
| *[[Schubert polynomials]], a generalization of Schur polynomials.
| |
| | |
| ==References==
| |
| *{{Cite book | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Symmetric functions and Hall polynomials | url=http://www.oup.com/uk/catalogue/?ci=9780198504504 | publisher=The Clarendon Press Oxford University Press | edition=2nd | series=Oxford Mathematical Monographs | isbn=978-0-19-853489-1 | id={{MathSciNet | id = 1354144}} | year=1995 | postscript=<!--None-->}}
| |
| *{{springer|id=s/s120040|title=Schur functions in algebraic combinatorics|first=Bruce E. |last=Sagan | authorlink=Bruce Sagan}}
| |
| *{{cite book | author=[[Bernd Sturmfels|Sturmfels, Bernd]] | title=Algorithms in Invariant Theory | location=New York | publisher=Springer | year=1993 | isbn=0-387-82445-6}}
| |
| | |
| [[Category:Homogeneous polynomials]]
| |
| [[Category:Invariant theory]]
| |
| [[Category:Representation theory of finite groups]]
| |
| [[Category:Symmetric functions]]
| |
| [[Category:Orthogonal polynomials]]
| |