Schubert polynomial

{{harvtxt|Lascoux|1995}} described the history of Schubert polynomials.

The Schubert polynomials $\backslash mathfrak\{S\}\_w$ are polynomials in the variables $x\_1,x\_2,\backslash ldots$ depending on an element $w$ of the infinite symmetric group $S\_\backslash infty$ of all permutations of $\backslash N$ fixing all but a finite number of elements. They form a basis for the polynomial ring $\backslash Z[x\_1,x\_2,\backslash ldots]$ in infinitely many variables.

The cohomology of the flag manifold $\backslash text\{Fl\}(m)$ is $\backslash Z[x\_1,\; x\_2,\backslash ldots,\; x\_m]/I,$ where $I$ is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial $\backslash mathfrak\{S\}\_w$ is the unique homogeneous polynomial of degree $\backslash ell(w)$ representing the Schubert cycle of $w$ in the cohomology of the flag manifold $\backslash text\{Fl\}(m)$ for all sufficiently large $m.${{citation needed|date=February 2014}}

• If $w\_0$ is the permutation of longest length in $S\_n$ then $\backslash mathfrak\{S\}\_\{w\_0\}\; =\; x\_1^\{n-1\}x\_2^\{n-2\}\; \backslash cdots\; x\_\{n-1\}^1$

• $\backslash partial\_i\; \backslash mathfrak\{S\}\_w\; =\; \backslash mathfrak\{S\}\_\{ws\_i\}$ if $w(i)\; >\; w(i+1)$, where $s\_i$ is the transposition $(i,\; i+1)$ and where $\backslash partial\_i$ is the divided difference operator taking $P$ to $(P\; -\; s\_iP)/(x\_i\; -\; x\_\{i+1\})$.

Schubert polynomials can be calculated recursively from these two properties. In particular, this implies that $\backslash mathfrak\{S\}\_w\; =\; \backslash partial\_\{w^\{-1\}w\_0\}\; x\_1^\{n-1\}x\_2^\{n-2\}\; \backslash cdots\; x\_\{n-1\}^1$.

Other properties are

• $\backslash mathfrak\{S\}\_\{id\}\; =\; 1$

• If $s\_i$ is the transposition $(i,i+1)$, then $\backslash mathfrak\{S\}\_\{s\_i\}\; =\; x\_1\; +\; \backslash cdots\; +\; x\_i$.

• If for all $i\; \backslash neq\; r$, then $\backslash mathfrak\{S\}\_w$ is the Schur polynomial $s\_\backslash lambda(x\_1,\backslash ldots,x\_r)$ where $\backslash lambda$ is the partition $(w(r)\; -\; r,\; \backslash ldots,\; w(2)\; -\; 2,\; w(1)\; -\; 1)$. In particular all Schur polynomials (of a finite number of variables) are Schubert polynomials.

• Schubert polynomials have positive coefficients. A conjectural rule for their coefficients was put forth by Richard P. Stanley, and proven in two papers, one by Sergey Fomin and Stanley and one by Sara Billey, William Jockusch, and Stanley.

• The Schubert polynomials can be seen as a generating function over certain combinatorial objects called pipe dreams or rc-graphs. These are in bijection with reduced Kogan faces, (introduced in the PhD thesis of Mikhail Kogan) which are special faces of the Gelfand-Tsetlin polytope.

• Schubert polynomials also can be written as a weighted sum of objects called bumpless pipe dreams.

As an example

:$\backslash mathfrak\{S\}\_\{51423\}(x)\; =\; x\_1\; x\_3^2\; x\_4\; x\_2^2+x\_1^2\; x\_3\; x\_4\; x\_2^2+x\_1^2\; x\_3^2\; x\_4\; x\_2.$

Since the Schubert polynomials form a $\backslash mathbb\; Z$-basis, there are unique coefficients $c^\{\backslash alpha\}\_\{\backslash beta\backslash gamma\}$

such that

:$\backslash mathfrak\{S\}\_\backslash beta\; \backslash mathfrak\{S\}\_\backslash gamma\; =\; \backslash sum\_\backslash alpha\; c^\{\backslash alpha\}\_\{\backslash beta\backslash gamma\}\; \backslash mathfrak\{S\}\_\backslash alpha.$

These can be seen as a generalization of the Littlewood−Richardson coefficients described by the Littlewood–Richardson rule.

For algebro-geometric reasons (Kleiman's transversality theorem of 1974), these coefficients are non-negative integers and it is an

outstanding problem in representation theory and combinatorics to give a combinatorial rule for these numbers.

Double Schubert polynomials $\backslash mathfrak\{S\}\_w(x\_1,x\_2,\backslash ldots,\; y\_1,y\_2,\backslash ldots)$ are polynomials in two infinite sets of variables, parameterized by an element w of the infinite symmetric group, that becomes the usual Schubert polynomials when all the variables $y\_i$ are $0$.

The double Schubert polynomial $\backslash mathfrak\{S\}\_w(x\_1,x\_2,\backslash ldots,\; y\_1,y\_2,\backslash ldots)$ are characterized by the properties

• $\backslash mathfrak\{S\}\_w(x\_1,x\_2,\backslash ldots,\; y\_1,y\_2,\backslash ldots)\; =\; \backslash prod\backslash limits\_\{i\; +\; j\; \backslash leq\; n\}\; (x\_i\; -\; y\_j)$ when $w$ is the permutation on $1,\backslash ldots,n$ of longest length.

• $\backslash partial\_i\; \backslash mathfrak\{S\}\_w\; =\; \backslash mathfrak\{S\}\_\{ws\_i\}$ if $w(i)\; >\; w(i+1).$

The double Schubert polynomials can also be defined as

:$\backslash mathfrak\{S\}\_w(x,y)\; =\backslash sum\_\{\; w=v^\{-1\}u\; \backslash text\{\; and\; \}\; \backslash ell(w)=\backslash ell(u)+\backslash ell(v)\; \}\; \backslash mathfrak\{S\}\_u(x)\; \backslash mathfrak\{S\}\_v(-y).$

{{harvtxt|Fomin|Gelfand|Postnikov|1997}} introduced quantum Schubert polynomials, that have the same relation to the (small) quantum cohomology of flag manifolds that ordinary Schubert polynomials have to the ordinary cohomology.

{{harvtxt|Fulton|1999}} introduced universal Schubert polynomials, that generalize classical and quantum Schubert polynomials. He also described universal double Schubert polynomials generalizing double Schubert polynomials.

• Stanley symmetric function
• Kostant polynomial
• Monk's formula gives the product of a linear Schubert polynomial and a Schubert polynomial.
• nil-Coxeter algebra

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Category:Representation theory

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