Bott–Samelson Resolution Overview

• Bott–Samelson resolution is a smooth, combinatorially explicit desingularization technique for Schubert varieties, constructed via iterated fiber products.
• It builds smooth varieties as towers of projective bundles using reduced word decompositions in the Weyl group, ensuring proper, birational maps.
• The method underpins applications in cohomology, K-theory, and representation theory by facilitating explicit pushforward computations and polynomial expansions.

The Bott–Samelson resolution is a key construction in the theory of singularities of Schubert varieties, providing a smooth, combinatorially explicit, equivariant desingularization with deep applications to algebraic geometry, representation theory, intersection theory, and cohomological invariants. It generalizes to multiple geometric settings, including generalized flag varieties, Grassmannians, and degeneracy loci, and underpins modern approaches to characteristic classes, Hecke algebras, and cohomology theories.

1. Classical and Generalized Construction

The classical Bott–Samelson variety is built from a reduced word for an element $w$ in the Weyl group $W$ associated with a reductive group $G$. Fixing a Borel subgroup $B\subset G$ and a reduced decomposition $w = s_{i_1}s_{i_2}\cdots s_{i_\ell}$ in simple reflections, define the Bott–Samelson variety as the iterated fiber product

$$Z_{(i_1,\ldots,i_\ell)} = P_{i_1} \times^B P_{i_2} \times^B \cdots \times^B P_{i_\ell} / B,$$

where each $P_{i_j}$ is the minimal parabolic containing $B$ corresponding to $\alpha_{i_j}$, and the associated quotient is via the right diagonal $B$-action (Rimanyi et al., 2019, Franco, 2023, Cibotaru, 2018).

The resolution map

$W$0

is given by multiplication $W$1 and is proper, birational, and $W$2-equivariant. The geometric structure is that of a smooth variety of dimension $W$3, constructed inductively as an iterated $W$4-bundle, with canonical sections corresponding to torus-fixed points and normal crossings boundary divisors (Rimanyi et al., 2019, 0903.3936, Silva, 2017).

Generalized Bott–Samelson varieties, as introduced by Perrin, allow for "good" decompositions $W$5, subject to minuscule and support compatibility conditions, yielding towers with locally trivial fibrations and birational maps to Schubert varieties in minuscule flag varieties (Brion et al., 2019).

2. Iterated Fiber Bundle Structure and Variants

The key geometric insight is the recursive construction as towers of projective bundles, with each stage corresponding to an addition of a simple reflection in the reduced expression. For each $W$6,

$W$7

and projections $W$8 are $W$9-fibrations. The choice of the base, such as a point, a minimal orbit, or a partial flag bundle, governs the variant. For orbit closures of square-zero upper-triangular matrices, the Bott–Samelson-type resolution begins with a smooth linear space and then applies the standard tower (Rudnicki et al., 2021).

Variants appear for relative Bott–Samelson varieties over a base scheme $G$0 and flag bundle $G$1, yielding resolutions of relative Schubert or Richardson varieties by fibered products of Schubert divisors (Li, 2020).

3. Exceptional Divisors, Normal Crossings, and Strictness

The resolution is exceptional in that its boundary divisor—the inverse image of the singular locus of the Schubert variety—admits a normal-crossing decomposition. For each projection $G$2, two canonical $G$3-fixed sections correspond to $G$4 and $G$5 in $G$6, and the images of the "infinity" sections produce the simple normal crossing divisors $G$7; their intersections correspond to Bott–Samelson varieties of subwords (Rimanyi et al., 2019, Rudnicki et al., 2021, Cibotaru, 2018).

Strictness—the property that the resolution is an isomorphism over the smooth locus—is combinatorially classified in type $G$8 by forbidden Bruhat patterns. For $G$9, outside an explicitly described family of permutations, one can always construct a strict Bott–Samelson resolution (Silva, 2017).

Bott–Samelson resolutions are not generally blow-ups: global functoriality and Cartier data may fail, but locally and for favorable cases (such as small dimensions), the exceptional loci may coincide with those of certain blow-up resolutions (Silva, 2017).

4. Functoriality and Applications to Cohomology, K-Theory, and Cobordism

The Bott–Samelson tower enables explicit description of pushforward classes in many generalized cohomology theories:

• In Levine–Morel algebraic cobordism $B\subset G$0, Bott–Samelson classes generate $B\subset G$1 as an $B\subset G$2-module (Lazars ring), with structure governed by universal formal group laws and divided-difference operators (0903.3936).
• In Chow or $B\subset G$3-theory, via specialization, the Bott–Samelson class expansions precisely recover double Schubert polynomials and Grothendieck polynomials (Hudson, 2013, Hudson, 2012).
• Connective $B\subset G$4-theory interpolation, with pushforwards determined by the double $B\subset G$5-polynomials of Fomin–Kirillov, unifies the previously separated cohomological and $B\subset G$6-theoretic formulas (Hudson, 2013).
• For equivariant cohomology and $B\subset G$7-theory, explicit localization formulas (Atiyah–Bott residues, GKM description) on the Bott–Samelson tower yield fixed-point computations of Schubert classes and their generalizations, including factorial Schur and Grothendieck representatives in Grassmannians (Oetjen, 2021).

For generalized flag bundles or relative settings, Bott–Samelson techniques provide the basis for Thom–Porteous formulae for degeneracy loci and intersection-theoretic classes (Hudson, 2013, Hudson, 2012).

5. Combinatorics, Hecke Algebras, and Schubert Calculus

The Bott–Samelson resolution is deeply entwined with the combinatorics of Weyl groups and Hecke algebras:

• The effective decomposition theorem realizes the derived pushforward of the resolution as a sum of intersection-cohomology complexes for Schubert subvarieties, with explicit multiplicity polynomials; this allows effective recursion for Kazhdan–Lusztig polynomials (Franco, 2023).
• The fiber geometry and cohomology support the construction of new Hecke algebra bases, transition matrices, and the study of connections to Kazhdan–Lusztig theory via characteristic cycles (Franco, 2023).
• In type $B\subset G$8, Bott–Samelson resolutions can be realized as quiver Grassmannians for appropriately defined quivers and representation parameters, providing a moduli-theoretic underpinning (Iezzi, 17 Feb 2025).
• Combinatorial structure is further governed by quiver diagrams of reduced words, peak decomposition, and explicit braid relations, particularly in generalized and minuscule settings (Brion et al., 2019).

6. Geometric and Representation-Theoretic Implications

The Bott–Samelson construction is instrumental for explicit geometric and representation-theoretic applications:

• In minuscule Schubert varieties and their small resolutions, key combinatorial theorems—equality of Weyl group stabilizers and root inequalities—control the geometry of minimal rational curves, yielding bijections between minimal rational curves on the resolution and lines on the Schubert variety (Brion et al., 2019).
• In cohomology, $B\subset G$9-theory, and cobordism, pushforwards via Bott–Samelson towers yield basis expansions in terms of Schubert (or Grothendieck) polynomials and support uniform Chevalley–Pieri and multiplication formulas (Li et al., 2020, 0903.3936).
• For orbit closures of square-zero matrices and related degeneracy loci, Bott–Samelson resolutions enable computation of Chern–Schwartz–MacPherson and motivic Chern classes via Demazure–Lusztig operators and equivariant localization (Rudnicki et al., 2021).

7. Alternative Resolutions and Further Developments

Bioriented-flag and Kempf–Laksov–type resolutions provide compelling alternatives, generalizing or complementing Bott–Samelson constructions:

• Bioriented-flag resolutions in the complete flag variety are geometric incarnations of certain Bott–Samelson resolutions, constructed without explicit choice of reduced words and via simple incidence relations (Cibotaru, 2018).
• In Grassmannians and degeneracy loci, Kempf–Laksov- and bioriented-flag resolutions extend the toolbox of available small resolutions, with connections to embedded desingularizations (Cibotaru, 2018).
• Relative Bott–Samelson constructions facilitate the desingularization of relative Richardson and Brill–Noether varieties, underpinning current research in geometric representation theory and algebraic geometry (Li, 2020).

The Bott–Samelson approach thus remains foundational across geometric representation theory, algebraic combinatorics, and modern intersection theory, continuously informing developments in Schubert calculus, equivariant theories, and moduli geometry.