Demazure Crystals in Algebra & Geometry

• Demazure crystals are extremal subcrystals of highest-weight crystals, encapsulating key combinatorial and representation-theoretic properties.
• They connect Kashiwara operators to Demazure modules and divided difference operators, yielding local recognition criteria and atomic decompositions.
• Applications include tableau models, key and Schubert polynomials, tensor product filtrations, and geometric parametrizations via polytopes.

A Demazure crystal is a specific extremal subcrystal of a highest-weight crystal for a symmetrizable Kac–Moody algebra, whose combinatorial and representation-theoretic properties encode Demazure modules, key polynomials, Schubert calculus, and a wide range of phenomena in algebraic combinatorics and geometry. Demazure crystals arise naturally by restricting crystal graphs to those elements obtainable via certain patterns of Kashiwara lowering operators corresponding to reduced words in the Weyl group, and admit deep connections to divided difference operators, atomic decompositions, and tensor product filtrations. Recent developments have extended the notion to flagged settings, Kohnert diagrams, reverse plane partitions, and have yielded fully local recognition criteria for unions and direct sums of Demazure crystals.

1. Definition and Characterization

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody Lie algebra with Weyl group $W$, weight lattice $P$, and dominant cone $P^+$. For $\lambda\in P^+$, the irreducible highest-weight crystal $B(\lambda)$ is equipped with raising and lowering crystal operators $e_i$, $f_i$ $(i\in I)$, weight maps, string functions, and satisfies the Kashiwara crystal axioms. Fixing $w\in W$ and any reduced word expression $w=s_{i_1}\dots s_{i_\ell}$, the Demazure crystal $B_w(\lambda)$ is the subcrystal generated by: $$B_w(\lambda) = \{ f_{i_1}^{a_1} \cdots f_{i_\ell}^{a_\ell}(b_\lambda) \mid a_j\ge0 \},$$ where $b_\lambda$ is the highest-weight element (Assaf et al., 22 Dec 2025). The construction is independent of choice of reduced word, and $B_w(\lambda)$ is stable under crystal operations.

Demazure crystals enjoy Kashiwara’s string property: the intersection of any $i$-string in $B(\lambda)$ with $B_w(\lambda)$ is either empty, the whole string, or the singleton highest element. This “extremality” property provides a local combinatorial signature distinguishing Demazure crystals.

The Demazure crystal $B_w(\lambda)$ is a crystal basis for the Demazure module $V_w(\lambda)$, and its character is the Demazure character: $$\mathrm{ch}\, B_w(\lambda) = \pi_{i_1}\cdots\pi_{i_\ell}(e^\lambda),$$ where $\pi_i$ are Demazure divided difference operators (Gibson, 2019).

2. Local and Atomic Structure

A major advance is the purely local characterization of single Demazure crystals and their unions (Assaf et al., 22 Dec 2025). For $X\subseteq B(\lambda)$, being a union of Demazure crystals is equivalent to satisfying two explicit local properties:

• Extremality (E): For every $i$-string, $S \cap X$ is empty, the whole string, or the highest string element only.
• Ideal subset (I): For extremal $x, y\in X$, if $x$ is obtained from $y$ via a sequence of raising operators, then corresponding lowering operations keep the result within $X$.

A principal extremal subset, with a unique maximal extremal weight, is a single Demazure crystal. General extremal subsets decompose into unions of Demazure crystals indexed by a lower ideal $J$ in Bruhat order: $X = B_J(\lambda) := \bigcup_{w\in J} B_w(\lambda).$ Each $B_w(\lambda)$ admits an atomic decomposition into Demazure atoms $A_w(\lambda)$ (elements in $B_w(\lambda)$ but not in any $B_v(\lambda)$ for $v\prec w$).

Type	Definition	Property
Extremal subset	$i$-string intersection is empty, singleton highest, or full	Union of Demazure crystals
Principal	Single maximal extremal	Single Demazure crystal
Atom	$$A_w(\lambda) := B_w(\lambda) \setminus \bigcup_{v<w} B_v(\lambda)$$	Minimal piece, disjoint

Atomic decompositions yield a filtration analogous to Schubert variety stratifications and underlie the structure of Polo modules (Assaf et al., 22 Dec 2025, Armon, 2023).

3. Crystal Models and Flagged Extensions

Type A admits explicit tableau models: key tableaux, flagged tableaux, reverse plane partitions, and factorization models (Wen, 2023, Kundu, 2023). Fix $\alpha$ a weak composition and $\varphi$ a flag, the space of $\varphi$-flagged key tableaux $SSKT(\alpha,\varphi)$ carries a crystal structure via bracketing, and corresponds precisely to a Demazure subcrystal with character the flagged key polynomial: $$\kappa_{(\alpha,\varphi)} = \sum_{T\in SSKT(\alpha,\varphi)} x^{wt(T)},$$ which is the output of explicit Demazure operator sequences applied to the ordinary key polynomial $\kappa_\alpha$ (Wen, 2023).

Flagged reverse plane partitions $R(\lambda/\mu,\Phi)$ also decompose combinatorially into a disjoint union of Demazure subcrystals via explicit insertion and bracketing algorithms, with associated generalized key positivity for Grothendieck polynomials (Kundu, 2023).

4. Tensor Products and Filtrations

Tensor products of Demazure crystals typically fail to be Demazure, unless a sharp local "hinge" criterion is satisfied (Assaf et al., 2022, Kouno, 2018):

• The tensor product $B_w(\lambda)\otimes B_u(\mu)$ is extremal (hence a direct sum of Demazure crystals) iff no broken $i$-hinge appears in the crystal graph for any $i$.
• Explicit global criteria are available: for every connected component to be Demazure, $[v]\in W^{W_\mu}$ (the minimal coset representative in the relevant parabolic) must hold (Kouno, 2018).

Applications include positivity of structure constants for key polynomial products, canonical decomposition of tensor squares, and filtration theorems for Polo-type modules (Assaf et al., 2022, Assaf et al., 22 Dec 2025).

5. Generalizations and Catalytic Models

Generalized Demazure and affine Demazure crystals are realized through successive application of Demazure and Dynkin automorphism operators in tensor products, admitting more complex combinatorial models such as katabolizable tableaux (DARK crystals) and product monomial crystals (Blasiak, 2020, Gibson, 2019). In the affine case, nonsymmetric Macdonald polynomials specialized at $t=0$ are precisely affine Demazure characters (Assaf et al., 2020, Assaf et al., 2019).

Crystals arising from Kohnert diagrams, flagged settings, or monomial models yield, via explicit local operators and rectification procedures, Demazure crystal components whose sum recovers central symmetric functions such as Schubert polynomials, key polynomials, Catalan functions, or nonsymmetric Macdonald polynomials (Assaf, 2020, Assaf et al., 2017, Blasiak et al., 2020).

6. Polyhedral and Geometric Perspective

Demazure crystals admit polyhedral parametrization: elements correspond to integer points in specific faces of string polytopes, Gelfand–Tsetlin polytopes, or Kogan faces (Fujita, 2020). The action of crystal operators matches combinatorial moves (mitosis, ladder moves) on these faces, and Schubert classes in cohomology correspond to explicit sums of these faces or crystal components.

This realizes geometric and representation-theoretic concepts (Schubert varieties, line bundle sections, extremal modules) directly in the combinatorics of crystal graphs and atomic decompositions, and connects to the theory of Bott–Samelson varieties and standard monomial theory.

7. Examples and Applications

Demazure crystals underpin much of algebraic combinatorics and geometry. For type A and a weak composition, key tableaux models, insertion algorithms, and Demazure operator recursions yield explicit formulas for key polynomials, Schubert polynomials, and their flagged variants (Wen, 2023, Assaf et al., 2017). Atomic decompositions enumerate Demazure atoms indexed by Bruhat order.

Tensor squares of Demazure crystals always contain a principal Demazure component corresponding to $B_w(2\lambda)$, and local criteria determine the decomposition into Demazure summands (Assaf et al., 2022). Affine and generalized Demazure crystals resolve positivity conjectures for k-Schur and Catalan functions (Blasiak et al., 2020, Blasiak, 2020). Interactions with quantum affine algebras, Kirillov–Reshetikhin crystals, and combinatorial models (rigged configurations, path models) reveal deep structural results and connections to physical partition function models (Schilling et al., 2011, Lenart et al., 2018, Yang, 5 Dec 2025).

In summary, Demazure crystals provide a combinatorial and representation-theoretic formalism for modeling submodules, filtrations, and polynomial families central to Lie theory, Schubert calculus, and algebraic combinatorics, with fully local recognition criteria and explicit combinatorial and geometric models now available for a wide range of generalizations and applications (Assaf et al., 22 Dec 2025, Wen, 2023, Assaf et al., 2022, Kouno, 2018, Blasiak, 2020, Gibson, 2019).