[Paper Review] Affine partitions and affine Grassmannians

Sara Billey, Stephen Α. Mitchell|ArXiv.org|Mar 25, 2008

Advanced Combinatorial Mathematics13 references4 citations

TL;DR

This paper establishes a bijection between elements of the quotient of an affine Weyl group modulo its finite Weyl group and a new family of combinatorial objects called affine partitions, which are colored partitions derived from canonical segment factorizations. The key contribution is a characterization of rationally smooth Schubert varieties in affine Grassmannians using these affine partitions and a generalized Young’s lattice that refines weak order and is refined by Bruhat order, with applications to partition identities and geometry in all classical and exceptional types, including new identities in non-type A settings.

ABSTRACT

We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type $A_{n}$, the affine colored partitions form another family of combinatorial objects in bijection with $n+1$-core partitions and $n$-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young's lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.

Motivation & Objective

• To establish a combinatorial bijection between minimal length coset representatives in the affine Weyl group quotient $\widetilde{W}/W$ and a new class of colored partitions called affine partitions.
• To unify and generalize existing bijections involving $k$-core, $k$-bounded, and skew partitions in type $A_n$ using the framework of affine partitions.
• To develop a generalized Young’s lattice on affine partitions that refines weak order and is refined by Bruhat order, enabling a new geometric-combinatorial characterization of rationally smooth Schubert varieties.
• To derive new partition identities from Bott’s formula via the bijection, extending known results from Bousquet-Mélou–Eriksson and Reiner to all types, including previously unknown identities in non-type $A$ settings.
• To provide a uniform, type-independent characterization of rationally smooth Schubert varieties in affine Grassmannians using the combinatorics of affine partitions and the generalized Young’s lattice.

Proposed method

• Construct a canonical factorization of elements in $\widetilde{W}^S$ into segments derived from the action of the finite Weyl group and automorphisms of the Dynkin diagram.
• Define affine partitions as colored partitions where each part is associated with a segment, preserving length via a weight-preserving map from $\widetilde{W}^S$ to the set of affine partitions.
• Introduce a generalized Young’s lattice on affine partitions that refines left weak order and is refined by Bruhat order, using segment-wise dominance and order relations.
• Use the bijection to translate geometric properties of Schubert varieties—such as rational smoothness—into combinatorial conditions on affine partitions.
• Leverage Bott’s formula for the Poincaré series of the affine Grassmannian to derive partition identities, with the generating function expressed as $P_{\widetilde{W}^S}(t) = \prod_{i=1}^n (1 - t^{e_i})^{-1}$, where $e_i$ are the exponents of $W$.
• Apply computer-assisted verification in certain cases to confirm combinatorial and order-theoretic properties, particularly in type $A_n$ and exceptional types.

Experimental results

Research questions

• RQ1 How can the minimal length coset representatives $\widetilde{W}^S$ of the affine Weyl group quotient $\widetilde{W}/W$ be bijectively mapped to a natural family of combinatorial objects?
• RQ2 What is the structure of a generalized Young’s lattice on affine partitions that refines weak order and is refined by Bruhat order?
• RQ3 How do the combinatorics of affine partitions reflect the geometry of Schubert varieties in affine Grassmannians, particularly rational smoothness?
• RQ4 What new partition identities arise from Bott’s formula via the affine partition bijection, and how do they relate to known identities in types $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, $G_2$?
• RQ5 Can the characterization of rationally smooth Schubert varieties be uniformly expressed in terms of affine partitions and their order-theoretic properties across all types?

Key findings

• A bijective correspondence is established between elements of $\widetilde{W}^S$ and affine partitions, where each element corresponds to a colored partition formed from canonical segment factorizations, preserving length and enabling combinatorial analysis.
• The generalized Young’s lattice on affine partitions refines left weak order and is refined by Bruhat order, providing a new poset structure that captures the hierarchy of Schubert varieties in affine Grassmannians.
• The paper provides a new characterization of rationally smooth Schubert varieties: $X_w$ is rationally smooth if and only if the lower order ideal $\{v \in \widetilde{W}^S : v \leq w\}$ is totally ordered, or $w$ is a closed parabolic orbit, or $w$ is spiral in type $A_n$, or $w$ is a specific element in type $B_3$.
• New partition identities are derived from Bott’s formula, extending known results from Bousquet-Mélou–Eriksson and Reiner to all types, with identities in types $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, $G_2$ appearing to be new.
• In type $A_n$, affine partitions provide a new combinatorial framework that unifies $n+1$-core partitions, $n$-bounded partitions, and skew shapes with no long hooks, offering a fresh perspective on these well-known families.
• The study confirms that the Poincaré series of the affine Grassmannian is given by $P_{\widetilde{W}^S}(t) = \prod_{i=1}^n (1 - t^{e_i})^{-1}$, where $e_i$ are the exponents of the finite Weyl group $W$, and this generating function is realized via the affine partition model.