Deodhar Geometry in Grassmannians

• Deodhar geometry is a combinatorial and geometric decomposition that refines Richardson varieties and Grassmannian projections using distinguished subexpressions and Go-diagrams.
• It employs network parametrizations and explicit affine-torus charts to control topology, positivity, and closure relations beyond traditional Schubert stratifications.
• Its applications span KP soliton theory, Wilson loop formulations in SYM, and Hecke-algebra structures, linking algebraic and combinatorial approaches in geometry.

Searching arXiv for recent and foundational papers on Deodhar geometry, Grassmannian decompositions, and related parametrizations. In current usage across recent work, Deodhar geometry is the geometry and combinatorics arising from Deodhar’s decomposition of Richardson varieties and from its projection to partial flag varieties, especially Grassmannians. Its basic objects are locally closed pieces indexed by distinguished subexpressions of reduced words; in the flag-variety setting these pieces are affine-torus charts, and in the Grassmannian they acquire concrete labels by Go-diagrams, network parametrizations, and Plücker vanishing data. A persistent feature of the subject is that the decomposition is finer than Schubert-, Richardson-, and positroid-level descriptions while remaining explicit enough to control topology, positivity, duality, and certain closure relations [(Speyer, 2023); (Talaska et al., 2012); (Singh, 10 Nov 2025)].

1. Origins in Richardson varieties and Bott–Samelson geometry

For $G/B$, with $G$ a semisimple or type-$A$ group and $W$ its Weyl group, Deodhar’s setting begins from Richardson varieties. In one standard notation,

$\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$

nonempty only when $v\le w$ in Bruhat order; in another,

$R_u^w=X_u\cap X^w.$

Open Richardson varieties are smooth, irreducible, affine, and satisfy

$\dim R_u^w=\ell(w)-\ell(u).$

Deodhar’s decomposition refines these varieties by fixing a reduced word and then indexing pieces by distinguished subexpressions, or equivalently by distinguished sequences $(v^0,\dots,v^a)$ in a Bott–Samelson model. Each Deodhar piece has the form

$$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$

with

$G$0

and the open Richardson variety is the union of those pieces with the prescribed final permutation (Speyer, 2023).

This geometry is closely tied to Bott–Samelson resolutions. For a reduced expression $G$1, Deodhar’s subexpression combinatorics gives the defect statistic

$G$2

and the polynomial

$G$3

A central geometric interpretation is that $G$4 is the Poincaré polynomial of the Bott–Samelson fiber over a Bruhat cell: $G$5 The derived pushforward of a Schubert resolution admits an effective decomposition-theorem expansion

$G$6

and the Laurent polynomials

$G$7

organize the support multiplicities. The recursive theorem computing $G$8 and the shifted Kazhdan–Lusztig data $G$9 from the fiber cohomology polynomials $A$0 makes Deodhar’s polynomials into effective input for Kazhdan–Lusztig theory and for new Hecke-algebra bases (Franco, 2023).

2. Projection to the Grassmannian and the role of Go-diagrams

In the Grassmannian, the decomposition is obtained by projection from the flag variety. With $A$1 and $A$2 Grassmannian,

$A$3

Every Deodhar component has simple topology: it is homeomorphic to $A$4. At the same time, the decomposition is not a stratification in general; closures of components are not unions of Deodhar components. This failure is one of the defining geometric subtleties of the subject (Singh, 10 Nov 2025).

The Grassmannian combinatorics is encoded by Go-diagrams. A Go-diagram is a filling of a Young diagram $A$5 by $A$6, interpreted through pipe dreams. The dictionary used across the Grassmannian literature is:

• $A$7: elbow compatible with the distinguished-subexpression rules,
• $A$8: crossing,
• $A$9: uncrossing.

A pipe dream is a Go-diagram exactly when the corresponding subexpression is distinguished, and a Go-diagram is an $W$0-diagram exactly when it is reduced, equivalently when the subexpression is positive distinguished. In the Grassmannian setting, unlike in the full flag variety, this indexing is independent of the chosen reduced word for $W$1. Go-diagrams therefore provide canonical combinatorial labels for Grassmannian Deodhar components, and Le-diagrams identify the totally nonnegative regime inside this broader decomposition (Talaska et al., 2012).

This Grassmannian viewpoint refines coarser decompositions. One formulation states that the Deodhar decomposition of the Grassmannian is a refinement of the Schubert, Richardson, and positroid stratifications. Another states that on the totally nonnegative Grassmannian, Deodhar pieces meet the nonnegative locus exactly when the indexing diagram is a Le-diagram; in that case the Deodhar piece agrees with the corresponding positroid cell. The resulting picture is that positroid combinatorics is the positive special case of a larger Deodhar-geometric structure (Marcott, 2018).

3. Parametrizations: Marsh–Rietsch factors, networks, bridge graphs, and pipe dreams

The original explicit parametrizations come from Marsh–Rietsch factorization on the flag side. For a distinguished subexpression, one builds

$W$2

and the map $W$3 is an isomorphism onto the corresponding Deodhar component. After projection to the Grassmannian, this yields explicit charts on projected Deodhar pieces, with lexicographically extremal nonzero Plücker coordinates determined by monomials in the $W$4-parameters (Kodama et al., 2012).

For Grassmannians, Talaska and Williams gave a direct network model from Go-diagrams, and a later network formulation made the parametrization fully explicit in terms of weighted graphs. From a Go-diagram $W$5, one constructs a weighted network $W$6 with internal vertices on $W$7- and $W$8-boxes, directed left and down, and weight matrix

$W$9

Letting the $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$0-weights vary over $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$1 and the black-stone weights over $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$2 produces

$\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$3

where $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$4 is the number of pluses and $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$5 the number of black stones. The Lindström–Gessel–Viennot lemma then turns minors of $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$6 into path counts, giving immediate Plücker vanishing and nonvanishing criteria and a minimal Plücker-coordinate characterization of the component (Talaska et al., 2012).

For positroid varieties, these projected Deodhar charts coincide with network charts coming from bridge graphs. Every Deodhar parametrization of a positroid variety corresponds to a bridge graph, and every bridge-graph parametrization agrees with some projected Deodhar parametrization. On both sides the resulting Grassmannian point is obtained by right-multiplication by elementary matrices $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$7, so the coincidence is not merely birational but formula-by-formula on the dense open chart (Karpman, 2014).

A newer refinement replaces the direct Go-network by a pipe-dream network based on restricted paths. A restricted path runs along pipes and may jump at $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$8- or $\mathcal{R}_{v,w}=B^+\dot w B^+\cap B^-\dot v B^+,$9-cells from a higher-labeled pipe to a lower-labeled pipe, with path weight equal to the product of the corresponding jump parameters. This gives

$v\le w$0

and, with

$v\le w$1

the resulting matrices realize exactly the points of $v\le w$2, uniquely. The product formula

$v\le w$3

makes the chart especially rigid. It yields transparent formulas for Grassmannian duality, where dual restricted paths give

$v\le w$4

once $v\le w$5, and it organizes Plücker coordinates by restricted diagrams rather than by unstructured path sums (Singh, 10 Nov 2025).

4. Positivity, solitons, Wilson loops, tableau keys, and Hecke-theoretic avatars

One of the most developed applications is to the KP equation. For $v\le w$6,

$v\le w$7

The positroid stratum of $v\le w$8 controls the unbounded line-solitons at $v\le w$9 and $R_u^w=X_u\cap X^w.$0, while the Deodhar component, encoded by a Go-diagram, controls the detailed contour plot for $R_u^w=X_u\cap X^w.$1. Black stones detect obstructions to total non-negativity and force singular behavior, and the regularity criterion is exact: $R_u^w=X_u\cap X^w.$2 In this setting Deodhar geometry is the mechanism that refines asymptotic positroid data into full contour-plot combinatorics (Kodama et al., 2012).

A different application appears in the geometry of tree-level Wilson loops in $R_u^w=X_u\cap X^w.$3 SYM. The delete-a-column map

$R_u^w=X_u\cap X^w.$4

has fibers over Wilson-loop positroid cells that decompose into unions of Deodhar components. Go-networks provide explicit coordinates on each Deodhar fiber and a diagrammatic procedure for reading the equations of a Deodhar component. This refinement detects when the lifted geometry leaves the positive Grassmannian and is also the combinatorial source of a non-orientability result for the associated real $R_u^w=X_u\cap X^w.$5-vector bundle over the Wilson-loop space (Agarwala et al., 2018).

Deodhar’s Bruhat-order lifts also appear in tableau combinatorics. In the symmetric-group/parabolic-coset setting, left keys of semistandard Young tableaux are obtained by repeated maximal Deodhar lifts, and right keys by repeated minimal Deodhar lifts. The explicit recursive rules

$R_u^w=X_u\cap X^w.$6

and

$R_u^w=X_u\cap X^w.$7

give constructive versions of the uniqueness of maximal and minimal lifts. This ties tableau keys to Bruhat geometry, Demazure characters, Standard Monomial Theory, and Lakshmibai–Seshadri paths (Kushwaha et al., 2023).

On the Hecke side, weighted Coxeter-group generalizations of Deodhar’s parabolic framework construct dual modules attached to $R_u^w=X_u\cap X^w.$8-graph ideals, recover the parabolic modules $R_u^w=X_u\cap X^w.$9 and $\dim R_u^w=\ell(w)-\ell(u).$0 when $\dim R_u^w=\ell(w)-\ell(u).$1, and extend Deodhar’s duality and inversion phenomena to unequal parameters. Together with the effective decomposition-theorem approach to $\dim R_u^w=\ell(w)-\ell(u).$2, this preserves Deodhar geometry’s original link to Kazhdan–Lusztig theory while broadening its algebraic range (Yin, 2015, Franco, 2023).

5. Closure phenomena, corrective flips, and the limits of diagrammatic classification

A central misconception addressed repeatedly in the literature is that Deodhar pieces should behave like a genuine stratification. They do not. In Lie type $\dim R_u^w=\ell(w)-\ell(u).$3, explicit examples show that closures of pieces need not be unions of pieces, and the same failure appears in Grassmannian formulations where closure containment is subtle and must be proved case by case (Speyer, 2023, Singh, 10 Nov 2025).

The combinatorial study of this failure is organized by corrective flips on $\dim R_u^w=\ell(w)-\ell(u).$4-diagrams. If a plus violates the distinguished property, a corrective flip swaps that plus with the white stone it violates, or with that stone’s uncrossing partner, then recolors stones so that black means uncrossing. These moves preserve the associated pair of permutations, preserve the numbers of black stones, white stones, and pluses, and terminate in a Go-diagram. They generalize Lam–Williams $\dim R_u^w=\ell(w)-\ell(u).$5-moves from reduced diagrams to non-reduced ones, but unlike the reduced case they need not produce a unique terminal diagram (Marcott, 2018).

This flip calculus yields codimension-one closure results. If one Go-diagram is obtained from another by replacing a crossing/uncrossing pair by pluses, or by replacing a suitable removable white stone by a plus, then recoloring and applying corrective flips produce a closure containment

$\dim R_u^w=\ell(w)-\ell(u).$6

with a one-dimensional difference in dimension. The same paper also proves that no reasonable forbidden-subdiagram characterization of Go-diagrams can exist: there is an injection from the set of valid Go-diagrams to the set of minimal forbidden subdiagrams. In place of finite pattern avoidance, the paper gives an inductive “partner” criterion: a diagram is a Go-diagram exactly when every black stone has a partner and every box with a partner is black (Marcott, 2018).

The pipe-dream parametrization sharpens this boundary theory. It proves a concrete closure statement for adjacent pipe labels: if $\dim R_u^w=\ell(w)-\ell(u).$7 has a crossing-uncrossing pair of pipes labeled $\dim R_u^w=\ell(w)-\ell(u).$8 and $\dim R_u^w=\ell(w)-\ell(u).$9, and $(v^0,\dots,v^a)$0 is obtained from $(v^0,\dots,v^a)$1 by changing that pair into two pluses, then

$(v^0,\dots,v^a)$2

provided $(v^0,\dots,v^a)$3 remains a Go-diagram. The proof uses the product formula for $(v^0,\dots,v^a)$4 and a degeneration procedure involving “excited factors” and “cooling sites.” The paper further formulates conjectures for nonadjacent labels and explicitly links this direction to earlier conjectures of Marcott (Singh, 10 Nov 2025).

6. Contemporary extensions: Deodhar tori, cluster structures, and nearby metric geometry

A survey-level synthesis places Deodhar geometry inside a larger theory of Richardson varieties, projected Richardson varieties, total positivity, and positroid varieties. In that formulation, Deodhar pieces are explicit affine-torus strata indexed by distinguished subexpressions, chamber minors recover the torus parameters, and in suitable reduced-word models rank conditions on submatrices give concrete descriptions of the pieces. The dense open Deodhar piece of an open Richardson variety is a torus $(v^0,\dots,v^a)$5, and the totally nonnegative part is obtained by imposing positivity on those torus coordinates (Speyer, 2023).

This torus viewpoint has been extended to braid varieties. For a double braid word $(v^0,\dots,v^a)$6, the Deodhar-side chart is the Deodhar torus

$(v^0,\dots,v^a)$7

defined by maximal-transversality conditions encoded by recursively defined Weyl-group elements $(v^0,\dots,v^a)$8. The complement of the torus is a union of Deodhar hypersurfaces, and the solid chamber minors $(v^0,\dots,v^a)$9 form a basis of characters of $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$0. The resulting Deodhar cluster variables $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$1 are characterized by vanishing to order $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$2 on the corresponding hypersurface and by nonvanishing on the others. An explicit comparison theorem identifies this Deodhar seed with the independently constructed weave seed on the corresponding braid variety: $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$3 Thus the Deodhar and weave cluster algebra structures coincide (Casals et al., 5 Aug 2025).

A nearby but indirect development studies metric algebraic geometry on $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$4 through the Grassmann distance degree. That work does not directly invoke Deodhar decompositions, but it gives explicit metric behavior for Schubert varieties, including a theorem that the $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$5 Schubert varieties

$$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$6

have GD degree $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$7, and it conjectures GD degree $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$8 for the remaining $$D_{\mathrm{seq}(v^0,\dots,v^a)}\cong (G_m)^{m_=}\times \mathbb A^{m_{\downarrow}},$$9. This suggests a metric layer on Schubert geometry compatible with the refined-cell perspective characteristic of Deodhar theory, although the connection is explicitly indirect (Friedman et al., 30 Jan 2026).

Across these directions, the common structure is stable: distinguished subexpressions provide the indexing, affine-torus charts provide the geometry, and diagrammatic or minor-theoretic models translate that geometry into explicit algebra. Deodhar geometry is therefore best viewed not as a single parametrization technique but as a family of precise refinements that turn Richardson and Grassmannian geometry into a manipulable combinatorial theory.