Peterson Schubert Calculus Overview

• Peterson Schubert calculus is the study of the cohomology of the Peterson variety, a special regular nilpotent Hessenberg variety in the flag variety.
• It employs four key ingredients: an affine paving for a geometric basis, restricted Schubert classes, explicit Monk and Giambelli formulas, and positive structure constants.
• It connects intersection theory, mixed Φ-Eulerian numbers, and the broader Peterson program linking quantum cohomology with affine and mirror-theoretic structures.

Peterson Schubert calculus is the Schubert-style study of the cohomology and equivariant cohomology of the Peterson variety, a special regular nilpotent Hessenberg variety inside the full flag variety $G/B$. In the papers under discussion, it consists of four tightly connected ingredients: a geometric basis coming from the affine paving by Peterson cells; cohomology classes obtained by restricting Schubert classes from $G/B$; multiplication rules such as Monk and Giambelli formulas; and positivity properties for the resulting structure constants. It also interfaces with intersection theory, mixed $\Phi$-Eulerian numbers, and the broader Peterson program relating Schubert calculus, quantum cohomology, and affine or mirror-theoretic structures (Insko et al., 2013, Drellich, 2013, Goldin et al., 2021, Gui et al., 7 Aug 2025).

1. Geometric setting and basic objects

Let $G$ be a complex reductive or semisimple algebraic group, $B$ a Borel subgroup, $T\subset B$ a maximal torus, and $W=N(T)/T$ the Weyl group. The full flag variety is

$G/B,$

with Bruhat decomposition

$G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$

and Schubert varieties

$X_w=\overline{C_w}=BwB/B.$

Their fundamental classes $G/B$0 form the standard Schubert basis of $G/B$1 (Insko et al., 2013).

A Hessenberg space is a subspace $G/B$2 such that

$G/B$3

Given $G/B$4, the Hessenberg variety is

$G/B$5

The Peterson variety is the regular nilpotent Hessenberg variety obtained from a regular nilpotent element

$G/B$6

and the Hessenberg space determined by the negative simple roots

$G/B$7

It is written as $G/B$8 in (Insko et al., 2013), as $G/B$9 in (Drellich, 2013), and as $\Phi$0 in (Goldin et al., 2021).

The basic combinatorics is indexed by subsets of the simple roots. If $\Phi$1, let $\Phi$2 be the parabolic subgroup generated by $\Phi$3, and $\Phi$4 its longest element. The $\Phi$5-fixed points of the Peterson variety are exactly the points $\Phi$6, equivalently

$\Phi$7

and

$\Phi$8

(Insko et al., 2013, Drellich, 2013).

A distinguished one-dimensional torus acts on the Peterson variety. In one formulation,

$\Phi$9

and in another, $G$0 is defined so that every simple root restricts to the same character $G$1. This reduction from $G$2 to a one-dimensional torus is essential because the full torus $G$3 does not preserve the Peterson variety, while $G$4 or $G$5 does (Drellich, 2013, Goldin et al., 2021, Goldin, 2023).

2. Affine paving and the homological framework

A foundational result is that regular nilpotent Hessenberg varieties admit affine pavings in all Lie types. For

$G$6

the intersection

$G$7

is either empty or isomorphic to affine space $G$8, where

$G$9

and

$B$0

Equivalently, nonemptiness is characterized by

$B$1

(Insko et al., 2013).

For the Peterson variety, the corresponding Peterson-Schubert cell is

$B$2

and its closure

$B$3

has dimension

$B$4

In the notation of (Goldin et al., 2021), the Peterson cells are

$B$5

with

$B$6

Thus the Peterson variety is paved by affine cells indexed by subsets of $B$7 (Insko et al., 2013, Goldin et al., 2021).

The affine paving has direct homological consequences. The closures of the cells

$B$8

freely generate

$B$9

and, in particular, the fundamental classes of Peterson cell closures form a basis of equivariant homology over $T\subset B$0 (Insko et al., 2013, Goldin et al., 2021, Goldin, 2023).

The inclusion of the Peterson variety into the flag variety induces

$T\subset B$1

A central theorem is that this map is injective. The proof uses the expansion of Peterson-Schubert classes in the Schubert basis and a triangularity statement for the coefficients, rather than the affine paving alone (Insko et al., 2013).

3. Peterson-Schubert classes and intersection-theoretic duality

The basic Schubert-calculus problem is to relate the homology classes of Peterson-Schubert varieties to the ordinary Schubert basis of $T\subset B$2. For a subset $T\subset B$3, (Insko et al., 2013) studies the pushforward

$T\subset B$4

and proves

$T\subset B$5

Moreover,

$T\subset B$6

where

$T\subset B$7

This identifies the leading Schubert coefficient as an intersection multiplicity (Insko et al., 2013).

The intersection-theoretic mechanism is precise. For each $T\subset B$8, one chooses an opposite Schubert variety $T\subset B$9 such that

$W=N(T)/T$0

and $W=N(T)/T$1 and $W=N(T)/T$2 intersect properly at $W=N(T)/T$3. The smoothness of $W=N(T)/T$4 in $W=N(T)/T$5 is controlled by Kumar’s criterion: it is smooth if and only if every connected block of $W=N(T)/T$6 is classically embeddable. This explains why intersection multiplicities can be subtle outside type $W=N(T)/T$7 (Insko et al., 2013).

In equivariant cohomology, the key classes are obtained by restricting Schubert classes from $W=N(T)/T$8. For each subset $W=N(T)/T$9, choose a Coxeter element $G/B,$0, and define

$G/B,$1

The central duality theorem states

$G/B,$2

where $G/B,$3 is the intersection multiplicity of the unique point

$G/B,$4

After normalization,

$G/B,$5

is an $G/B,$6-basis of $G/B,$7 (Goldin et al., 2021).

This duality gives a geometric meaning to Peterson Schubert classes. They are not merely formal pullbacks; up to the multiplicity $G/B,$8, they are dual to the fundamental classes of the closures of Peterson cells. A plausible implication is that the choice of basis in Peterson Schubert calculus is forced by the geometry of the affine paving together with the ambient Schubert calculus of $G/B,$9 (Goldin et al., 2021, Goldin, 2023).

4. Multiplication rules: Monk, Giambelli, and all structure constants

A decisive step is the construction of explicit multiplication rules. In all Lie types, the Peterson Schubert basis in $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$0 may be indexed by subsets $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$1 via special Weyl group elements $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$2, and the degree-one classes $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$3 are the ring generators (Drellich, 2013).

The all–Lie-type Monk formula has the form

$G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$4

with explicit coefficients

$G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$5

All other coefficients vanish, and the coefficients lie in $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$6. The paper notes that non-integral structure constants can occur in types $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$7 and $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$8, and conjectures these are the only cases where non-integrality appears (Drellich, 2013).

The all–Lie-type Giambelli formula expresses every Peterson Schubert class as a scalar multiple of a product of the degree-one generators. For connected $G/B=\bigsqcup_{w\in W} C_w,\qquad C_w=BwB/B,$9,

$X_w=\overline{C_w}=BwB/B.$0

in types $X_w=\overline{C_w}=BwB/B.$1, and the same formula holds in types $X_w=\overline{C_w}=BwB/B.$2 and $X_w=\overline{C_w}=BwB/B.$3, with a more delicate proof. Uniformly,

$X_w=\overline{C_w}=BwB/B.$4

where $X_w=\overline{C_w}=BwB/B.$5 is the number of reduced words for $X_w=\overline{C_w}=BwB/B.$6. For disconnected

$X_w=\overline{C_w}=BwB/B.$7

the constant factors multiplicatively: $X_w=\overline{C_w}=BwB/B.$8 This makes the degree-one Peterson Schubert classes the exact analogue of special Schubert classes (Drellich, 2013).

A more recent development is an explicit, positive, type-uniform formula for all equivariant structure constants in arbitrary Lie type. In the Harada–Horiguchi–Masuda presentation,

$X_w=\overline{C_w}=BwB/B.$9

the equivariant Peterson Schubert class satisfies

$G/B$00

If

$G/B$01

then

$G/B$02

and the coefficients $G/B$03 are obtained by explicit matrix products built from inverse Cartan matrices $G/B$04. This reduces Peterson Schubert multiplication to closed linear-algebraic formulas depending only on the Cartan matrix (Gui et al., 7 Aug 2025).

5. Positivity and combinatorial models

A major theme is positivity. In $G/B$05, the multiplication

$G/B$06

has coefficients

$G/B$07

Similarly, if

$G/B$08

then

$G/B$09

These are Peterson analogues of Graham positivity for ordinary Schubert calculus on $G/B$10 (Goldin et al., 2021, Goldin, 2023).

The geometric explanation is that the Peterson basis is obtained from pullbacks of Schubert classes and is dual, up to multiplicity, to the affine paving basis. Since the positivity theorem for $G/B$11 is available in the ambient flag variety, positivity for the Peterson variety is deduced by combining that ambient positivity with the duality theorem for Peterson cells (Goldin et al., 2021, Goldin, 2023).

In type $G/B$12, this positivity becomes fully combinatorial. The basis

$G/B$13

forms an $G/B$14-module basis of $G/B$15, with

$G/B$16

and for consecutive $G/B$17,

$G/B$18

Products expand as

$G/B$19

where each nonzero coefficient is a nonnegative, integral multiple of a power of $G/B$20: $G/B$21 The nonvanishing criterion is: $G/B$22 if and only if $G/B$23 and for every maximal consecutive component $G/B$24 of $G/B$25,

$G/B$26

For consecutive nonempty $G/B$27, the coefficients are given by explicit multinomial formulas (2004.05959).

A geometry-first type $G/B$28 model replaces abstract basis elements by subvarieties

$G/B$29

with

$G/B$30

The corresponding cohomology classes

$G/B$31

form a $G/B$32-basis of $G/B$33, are dual to $G/B$34, and coincide with the Harada–Tymoczko basis: $G/B$35 The structure constants satisfy

$G/B$36

and admit a manifestly positive formula

$G/B$37

in terms of left-right diagrams (Abe et al., 2021).

6. Extensions: mixed $G/B$38-Eulerian numbers and the broader Peterson program

Peterson Schubert calculus also computes mixed $G/B$39-Eulerian numbers. For a root system $G/B$40, the mixed numbers are the coefficients in the volume polynomial

$G/B$41

They can be realized as Peterson intersections: $G/B$42 and

$G/B$43

In type $G/B$44, this reproduces the Berget–Spink–Tseng computation, while in arbitrary Lie type the recursive computation is governed by Peterson Monk/Giambelli rules (Horiguchi, 2021).

The type-uniform matrix formula for Peterson structure constants immediately yields a type-uniform formula for mixed $G/B$45-Eulerian numbers: $G/B$46 This ties weight-polytope volumes directly to Peterson multiplication (Gui et al., 7 Aug 2025).

The subject also sits inside a broader Peterson program. One branch identifies the spectrum of $G/B$47 with strata of the Peterson variety in the Langlands dual flag variety. Another branch studies smooth Schubert varieties whose quantum cohomology appears to be governed by Peterson-type or mirror-theoretic structures. For the smooth Schubert divisor

$G/B$48

the paper (Li et al., 22 Sep 2025) gives a Borel-type presentation of $G/B$49, a quantum Chevalley formula, and the statement that the quantum Schubert polynomials for $G/B$50 are the same as those for $G/B$51. The paper explicitly says that its presentation was predicted in the extended Peterson perspective of Li–Rietsch–Yang (Li et al., 22 Sep 2025).

A related affine direction is the Peterson–Lam–Shimozono theorem, interpreted as an affine analogue of the quantum Chevalley formula. It identifies $G/B$52-equivariant quantum Schubert calculus on $G/B$53 with $G/B$54-equivariant affine Schubert calculus on the affine Grassmannian $G/B$55, via a map sending

$G/B$56

This places Peterson Schubert calculus within a larger network linking flag varieties, affine Schubert geometry, and quantum multiplication (Chow, 2021).

Taken together, these developments present Peterson Schubert calculus as a subvariety analogue of classical Schubert calculus on $G/B$57: it has a geometric basis from cell closures, a cohomological basis obtained from restricted Schubert classes, explicit multiplication rules, positive structure constants, and concrete interfaces with intersection theory, Lie-type combinatorics, and quantum geometry (Insko et al., 2013, Drellich, 2013, Goldin et al., 2021, Gui et al., 7 Aug 2025).