[Paper Review] A Pieri-type Formula for the Equivariant Cohomology of the Flag Manifold

Shawn Robinson|ArXiv.org|Sep 21, 2000

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper establishes a Pieri-type formula for the equivariant cohomology of the flag manifold $G/B$ with $G = SL_n(\mathbb{C})$, providing a combinatorial rule to compute the structure constants in the product $\xi^c \xi^w = \sum p_{c,w}^u \xi^u$, where $c = c[k,m]$ is a product of consecutive simple reflections. The key result expresses these coefficients as evaluations of certain polynomials $\xi^{c[k-p,m-p]}(v_{[u,w,k]})$ over a set of permutations $u$ satisfying a special $k$-superiority condition, generalizing classical Pieri rules to the equivariant setting with explicit positivity.

ABSTRACT

We prove an explicit combinatorial formula for certain structure constants of the T-equivariant cohomology of the flag manifold SLn/B. Our result generalizes the Pieri-type formula in ordinary cohomology proved by Sottile in 1996. Our result also gives a Pieri-type formula for the double Schubert polynomials introduced by Lascoux and Schutzenberger.

Motivation & Objective

• To generalize the classical Pieri formula to the $T$-equivariant cohomology of the flag variety $G/B$ for $G = SL_n(\mathbb{C})$.
• To provide a combinatorial formula for the structure constants $p_{c,w}^u$ in the product $\xi^c \xi^w = \sum p_{c,w}^u \xi^u$, where $\xi^w$ are equivariant Schubert classes.
• To demonstrate that these structure constants, which are polynomials in $S(\mathfrak{h}^*)$, are explicitly positive via evaluation of specific polynomials.

Proposed method

• Define the equivariant Schubert basis $\{\xi^w\}_{w \in W}$ via functions $\xi^w: W \to S(\mathfrak{h}^*)$ constructed by Kostant and Kumar.
• Introduce the notion of $u$ being special $k$-superior to $w$ of degree $p$, characterized by conditions on values, positions, and length increase.
• Formulate the main result: $\xi^c \xi^w = \sum_{0 \leq p \leq m, \, u \in S_{k,p}(w)} \xi^{c[k-p,m-p]}(v_{[u,w,k]}) \xi^u$, where $c = c[k,m]$ is a product of $m$ consecutive simple reflections.
• Use induction on $k$ and recursive decomposition of $\xi^{c[k,m]}$ into $\xi^{c[k-1,m]} + \xi^{c[k-1,m-1]}(\xi^{s_k} - \xi^{s_{k-1}} + L_{k-m+1} - L_k)$.
• Leverage lemmas on the structure of $S_{k,p}(w)$, polynomial evaluations $\xi^w(v)$, and the behavior of $\xi^w$ under multiplication by $\xi^{s_k}$.
• Verify positivity and correctness through recursive application of structure constants and polynomial identities, including the use of transition maps $v_{[u,w,k]}$.

Experimental results

Research questions

• RQ1 How can the classical Pieri formula be extended to the equivariant cohomology of the flag manifold $SL_n(\mathbb{C})/B$?
• RQ2 What combinatorial conditions on permutations $u$ ensure non-zero structure constants $p_{c,w}^u$ in the product $\xi^c \xi^w$?
• RQ3 Can the structure constants $p_{c,w}^u$ be expressed explicitly as evaluations of polynomials in $S(\mathfrak{h}^*)$?
• RQ4 Does the formula exhibit the known positivity of equivariant structure constants, and if so, how is this made manifest?
• RQ5 How does the formula reduce to known results, such as the Pieri rule for Schubert polynomials, in special cases?

Key findings

• The paper provides a complete Pieri-type formula for $\xi^c \xi^w$ in $H_T^*(SL_n(\mathbb{C})/B)$, where $c = c[k,m]$ is a product of $m$ consecutive simple reflections.
• The structure constants are given by $p_{c,w}^u = \xi^{c[k-p,m-p]}(v_{[u,w,k]})$ for $u \in S_{k,p}(w)$, with $S_{k,p}(w)$ defined by $k$-superiority conditions.
• When $p = m$, the formula reduces to the classical Pieri rule for Schubert polynomials, recovering the result of Lascoux and Schützenberger.
• The positivity of the structure constants is explicitly demonstrated through the evaluation of polynomials $\xi^{c[k-p,m-p]}(v_{[u,w,k]})$, which are shown to be non-negative under the given conditions.
• The formula is proven via induction on $k$, using recursive decomposition of $\xi^{c[k,m]}$ and detailed analysis of the coefficient contributions from $\xi^{c[k-1,m]}$, $\xi^{c[k-1,m-1]}$, and transition terms involving transpositions $t_{kq}$ and $t_{qk}$.
• The proof establishes that coefficients vanish when $u \notin S_{k,p}(w)$, even in cases where $u$ appears in $S_{k-1,p}(w)$, by showing cancellation between terms involving $ut_{kq}$ and $ut_{q'k}$.