[Paper Review] From quantum Schubert polynomials to k-Schur functions via the Toda lattice

Thomas Lam, Mark Shimozono|arXiv (Cornell University)|Oct 19, 2010

Advanced Combinatorial Mathematics13 references3 citations

TL;DR

This paper establishes a direct algebraic correspondence between quantum Schubert polynomials and $k$-Schur functions via a rational substitution derived from Kostant's solution to the Toda lattice and Peterson's isomorphism. The key result shows that quantum Schubert polynomials map to $k$-Schur functions through a substitution involving ratios of symmetric functions associated with rectangular partitions, with denominators corresponding to the descent set of the permutation.

ABSTRACT

We show that Lapointe-Lascoux-Morse k-Schur functions (at t=1) and Fomin-Gelfand-Postnikov quantum Schubert polynomials can be obtained from each other by a rational substitution. This is based upon Kostant's solution of the Toda lattice and Peterson's work on quantum Schubert calculus.

Motivation & Objective

• To establish a precise algebraic correspondence between quantum Schubert polynomials and $k$-Schur functions.
• To show that the transition between these two polynomial bases arises naturally from the solution of the Toda lattice.
• To provide an explicit rational substitution map $\Phi$ that transforms quantum Schubert polynomials into $k$-Schur functions.
• To unify quantum cohomology of the flag manifold with homology of the affine Grassmannian via Toda lattice structures.
• To offer a combinatorial realization of $k$-Schur functions using quantum Schubert polynomials and symmetric function ratios.

Proposed method

• Utilize Kostant's solution of the nilpotent Toda lattice to relate the coordinate ring of the Toda lattice to quantum cohomology of the flag manifold.
• Apply Peterson's isomorphism between the homology of the affine Grassmannian and the centralizer of a principal nilpotent element in $PGL(n)$.
• Define a substitution map $\Phi$ from quantum Schubert polynomials to symmetric functions in $\Lambda_{(n)}[s_{R_i}^{-1}]$ via ratios $s_{R'_i}/s_{R_i}$ for $x_1+\cdots+x_i$ and $s_{R_{i-1}}s_{R_{i+1}}/s_{R_i}^2$ for $q_i$.
• Leverage Fomin-Gelfand-Postnikov's quantum Schubert polynomials as representatives of quantum Schubert classes in $QH^*(\mathrm{Fl}_n)$.
• Use Lam's realization of $k$-Schur functions as affine Schubert classes in $H_*(\mathrm{Gr})$ to connect the two structures.
• Verify the correspondence via explicit computation on examples, such as $w = s_1s_2s_1$, showing agreement with $s_1^{(2)} = h_1$.

Experimental results

Research questions

• RQ1 How are quantum Schubert polynomials related to $k$-Schur functions through algebraic structures?
• RQ2 Can the Toda lattice provide a unified framework for quantum Schubert calculus and affine Schubert calculus?
• RQ3 What rational substitution transforms quantum Schubert polynomials into $k$-Schur functions?
• RQ4 How does the descent set of a permutation appear in the denominator of the $k$-Schur function image under $\Phi$?
• RQ5 Is there a conceptual link between the Toda lattice and Schubert calculus in both quantum and affine settings?

Key findings

• The map $\Phi$ sends quantum Schubert polynomials $\mathfrak{S}_w^q$ to $\frac{s_{\lambda(w)}^{(k)}}{\prod_{i \in \mathrm{Des}(w)} s_{R_i}}$, establishing a direct correspondence.
• For $w = s_1s_2s_1$ in $S_3$, $\Phi(\mathfrak{S}_w^q) = \frac{h_1}{e_2 h_2} = s_1^{(2)}$, confirming the formula with $\lambda(w) = (1)$ and $\mathrm{Des}(w) = \{1,2\}$.
• The substitution $\Phi$ is derived from Kostant's solution of the Toda lattice and Peterson's isomorphism, linking quantum cohomology and affine Grassmannian homology.
• The denominator in the result corresponds to the descent set of $w$, with each $s_{R_i}$ arising from the Toda lattice structure.
• The construction relies on the factorization of $k$-Schur functions via Lapointe-Morse's result, which allows full reconstruction of $s_\lambda^{(k)}$ from the image of $\Phi$.
• The map $\Phi$ is equivalent to substituting $e_i(m)$ with $\frac{h_i^\perp \cdot S_{R_m}}{S_{R_m}}$ in the Schubert polynomial expansion, yielding the same $k$-Schur function image.