---
title: Peterson Schubert Calculus Overview
url: https://www.emergentmind.com/topics/peterson-schubert-calculus
type: topic
---

# Peterson Schubert Calculus Overview

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 [1309.0484], [1311.3014], [2106.10372], [2508.05457].

## 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 \([X_w]\) form the standard Schubert basis of \(H_*(G/B,\mathbb Z)\) [1309.0484].

A Hessenberg space is a subspace \(H\subset \mathfrak g\) such that
\[
\mathfrak b\subset H,\qquad [H,\mathfrak b]\subset H.
\]
Given \(X\in\mathfrak g\), the Hessenberg variety is
\[
H(X,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})(X)\in H\}.
\]
The Peterson variety is the regular nilpotent Hessenberg variety obtained from a regular nilpotent element
\[
X_0=\sum_{\alpha_i\in \Delta} E_{\alpha_i}
\]
and the Hessenberg space determined by the negative simple roots
\[
M_H=\Delta^{-}.
\]
It is written as \(H(X,\mathfrak h)\) in [1309.0484], as \(\mathrm{Pet}\) in [1311.3014], and as \(P\) in [2106.10372].

The basic combinatorics is indexed by subsets of the simple roots. If \(J\subset \Delta\), let \(W_J\) be the parabolic subgroup generated by \(J\), and \(w_J\) its longest element. The \(S\)-fixed points of the Peterson variety are exactly the points \(w_JB\), equivalently
\[
(\mathrm{Pet})^{S^1}=\mathrm{Pet}\cap (G/B)^T,
\]
and
\[
wB\in(\mathrm{Pet})^{S^1}\iff w=w_J\ \text{for some }J\subset\Delta
\]
[1309.0484], [1311.3014].

A distinguished one-dimensional torus acts on the Peterson variety. In one formulation,
\[
S^1=\sigma^{-1}\{(c,c,\dots,c):c\in\mathbb C^*\},
\]
and in another, \(S\subset T\) is defined so that every simple root restricts to the same character \(t\). This reduction from \(T\) to a one-dimensional torus is essential because the full torus \(T\) does not preserve the Peterson variety, while \(S\) or \(S^1\) does [1311.3014], [2106.10372], [2306.14391].

## 2. Affine paving and the homological framework

A foundational result is that regular nilpotent Hessenberg varieties admit affine pavings in all Lie types. For
\[
X=\sum_{\alpha_i\in\Delta}E_{\alpha_i},
\]
the intersection
\[
C_w\cap H(X,H)
\]
is either empty or isomorphic to affine space \(\mathbb C^d\), where
\[
d=\bigl|N(w)\cap M_H\bigr|
\]
and
\[
N(w)=\{\alpha\in \Phi^+\mid w^{-1}\alpha<0\}.
\]
Equivalently, nonemptiness is characterized by
\[
w^{-1}(\alpha_i)\in M_H \quad \text{for all simple roots } \alpha_i
\]
[1309.0484].

For the Peterson variety, the corresponding Peterson-Schubert cell is
\[
C_{w_J,\mathfrak h}=C_{w_J}\cap H(X,\mathfrak h),
\]
and its closure
\[
X_{w_J,\mathfrak h}=\overline{C_{w_J,\mathfrak h}}
\]
has dimension
\[
\dim X_{w_J,\mathfrak h}=|J|.
\]
In the notation of [2106.10372], the Peterson cells are
\[
P_I^\circ:=P\cap Bw_IB/B,
\qquad
P_I=\overline{P_I^\circ},
\]
with
\[
P_I^\circ\simeq \mathbb C^{|I|},
\qquad
P=\bigsqcup_{I\subset\Delta} P_I^\circ,
\qquad
P_I=\bigsqcup_{J\subset I} P_J^\circ.
\]
Thus the Peterson variety is paved by affine cells indexed by subsets of \(\Delta\) [1309.0484], [2106.10372].

The affine paving has direct homological consequences. The closures of the cells
\[
\overline{C_w\cap H(X,H)}
\]
freely generate
\[
H_*(H(X,H),\mathbb Z),
\]
and, in particular, the fundamental classes of Peterson cell closures form a basis of equivariant homology over \(H_S^*(\mathrm{pt})\) [1309.0484], [2106.10372], [2306.14391].

The inclusion of the Peterson variety into the flag variety induces
\[
i_*:H_*(H(X,\mathfrak h),\mathbb Z)\to H_*(G/B,\mathbb Z).
\]
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 [1309.0484].

## 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 \(G/B\). For a subset \(J\subset \Delta\), [1309.0484] studies the pushforward
\[
i_*\bigl([X_{w_J,\mathfrak h}]\bigr)=\sum_{K} a_{J,K}\,[X_{w_K}],
\]
and proves
\[
a_{J,J}\neq 0,
\qquad
a_{J,K}=0 \text{ whenever } K\neq J \text{ and } |K|=|J|.
\]
Moreover,
\[
a_{J,J}=b_J
\]
where
\[
[X_{w_J,\mathfrak h}]\cdot [X^{u_J}] = b_J [w_JB], \qquad b_J>0.
\]
This identifies the leading Schubert coefficient as an intersection multiplicity [1309.0484].

The intersection-theoretic mechanism is precise. For each \(J\), one chooses an opposite Schubert variety \(X^{u_J}\) such that
\[
X_{w_J,\mathfrak h}\cap X^{u_J}=\{w_JB\},
\]
and \(X_{w_J,\mathfrak h}\) and \(X^{u_J}\) intersect properly at \(w_JB\). The smoothness of \(w_JB\) in \(X^{u_J}\) is controlled by Kumar’s criterion: it is smooth if and only if every connected block of \(J\) is classically embeddable. This explains why intersection multiplicities can be subtle outside type \(A\) [1309.0484].

In equivariant cohomology, the key classes are obtained by restricting Schubert classes from \(G/B\). For each subset \(I\subset\Delta\), choose a Coxeter element \(v_I\in W\), and define
\[
p_I:=\iota^*(\sigma_{v_I})\in H_S^{2|I|}(P).
\]
The central duality theorem states
\[
\left\langle \iota^*\sigma_{v_I}, [P_J]_S\right\rangle = m(v_I)\,\delta_{I,J},
\]
where \(m(v_I)\in \mathbb Z_{>0}\) is the intersection multiplicity of the unique point
\[
X^{v_I}\cap P_I=\{w_I\}.
\]
After normalization,
\[
\left\{\frac{p_I}{m(v_I)}\right\}_{I\subset\Delta}
\]
is an \(H_S^*(pt)\)-basis of \(H_S^*(P)\) [2106.10372].

This duality gives a geometric meaning to Peterson Schubert classes. They are not merely formal pullbacks; up to the multiplicity \(m(v_I)\), 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 \(G/B\) [2106.10372], [2306.14391].

## 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 \(H_{S^1}^*(\mathrm{Pet})\) may be indexed by subsets \(K\subset\Delta\) via special Weyl group elements \(v_K\), and the degree-one classes \(p_{s_i}\) are the ring generators [1311.3014].

The all–Lie-type Monk formula has the form
\[
p_{s_i}\,p_{v_K} = p_{s_i}(w_K)\,p_{v_K} + \sum_{\substack{J\supset K\ |J|=|K|+1} c_{i,K}^{\,J}\,p_{v_J},
\]
with explicit coefficients
\[
c_{i,K}^{\,J} = \frac{\big(p_{s_i}(w_J)-p_{s_i}(w_K)\big)\,p_{v_K}(w_J)}{p_{v_J}(w_J)}.
\]
All other coefficients vanish, and the coefficients lie in \(\mathbb Q_{\ge 0}\). The paper notes that non-integral structure constants can occur in types \(D\) and \(E\), and conjectures these are the only cases where non-integrality appears [1311.3014].

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 \(K\),
\[
|K|!\,p_{v_K}=\prod_{\alpha_i\in K} p_{s_i}
\]
in types \(A_n,B_n,C_n,F_4,G_2\), and the same formula holds in types \(D_n\) and \(E_n\), with a more delicate proof. Uniformly,
\[
|K|!\,|R(v_K)|\; p_{v_K}=\prod_{\alpha_i\in K} p_{s_i},
\]
where \(|R(v_K)|\) is the number of reduced words for \(v_K\). For disconnected
\[
K=K_1\sqcup\cdots\sqcup K_m,
\]
the constant factors multiplicatively:
\[
C_K\,p_{v_K}=\prod_{\alpha_i\in K}p_{s_i},
\qquad
C_K=\prod_{r=1}^m \frac{|K_r|!}{|R(v_{K_r})|}.
\]
This makes the degree-one Peterson Schubert classes the exact analogue of special Schubert classes [1311.3014].

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,
\[
H_S^*(\Pet_G;\mathbb Q)\cong \frac{\mathbb Q[\varpi_1,\dots,\varpi_n,t]} {\left(\sum_{j=1}^n\langle \alpha_i,\alpha_j\rangle \varpi_i\varpi_j-2t\,\varpi_i\mid 1\le i\le n\right)},
\]
the equivariant Peterson Schubert class satisfies
\[
p_I \longleftrightarrow \frac{\det(C_I)}{|W_I|}\,\varpi_I,
\qquad
\varpi_I:=\prod_{i\in I}\varpi_i.
\]
If
\[
p_I\cdot p_J=\sum_{K\subset\Delta} c_{I,J}^K\,p_K,
\]
then
\[
c_{I,J}^K= \frac{\det(C_I)\det(C_J)|W_K|}{|W_I||W_J|\det(C_K)}\,d_{I,J}^K,
\]
and the coefficients \(d_{I,J}^K\) are obtained by explicit matrix products built from inverse Cartan matrices \(C_K^{-1}\). This reduces Peterson Schubert multiplication to closed linear-algebraic formulas depending only on the Cartan matrix [2508.05457].

## 5. Positivity and combinatorial models

A major theme is positivity. In \(H_S^*(P)\), the multiplication
\[
p_I\cdot p_J=\sum_K c_{I,J}^K\, p_K
\]
has coefficients
\[
c_{I,J}^K\in \mathbb Z_{\ge 0}[t].
\]
Similarly, if
\[
\iota_*[P_I]_S=\sum_{v\in W} c_I^v [X_v]_S,
\]
then
\[
c_I^v\in \mathbb Z_{\ge 0}[t].
\]
These are Peterson analogues of Graham positivity for ordinary Schubert calculus on \(G/B\) [2106.10372], [2306.14391].

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 \(G/B\) 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 [2106.10372], [2306.14391].

In type \(A\), this positivity becomes fully combinatorial. The basis
\[
\{p_A\}_{A\subseteq [n-1]}
\]
forms an \(H_S^*\)-module basis of \(H_S^*(Y)\), with
\[
\deg(p_A)=2|A|,
\qquad
p_A|_{w_C}=0 \quad \text{unless } A\subseteq C,
\]
and for consecutive \(A\),
\[
p_A|_{w_A}=|A|!\, t^{|A|}.
\]
Products expand as
\[
p_A p_B = \sum_{C\subseteq \{1,\ldots,n-1\}} b_{A,B}^C\, p_C,
\]
where each nonzero coefficient is a nonnegative, integral multiple of a power of \(t\):
\[
b_{A,B}^C = a\, t^d,
\qquad
a>0,
\qquad
d=|A|+|B|-|C|.
\]
The nonvanishing criterion is:
\[
b_{A,B}^C\neq 0
\]
if and only if \(A\cup B\subseteq C\) and for every maximal consecutive component \(C_k\) of \(C\),
\[
|C_k|\le |C_k\cap A|+|C_k\cap B|.
\]
For consecutive nonempty \(A,B,C\), the coefficients are given by explicit multinomial formulas [2004.05959].

A geometry-first type \(A\) model replaces abstract basis elements by subvarieties
\[
X_J := X_{w_J}\cap \mathrm{Pet}_n,
\qquad
Q_J := Q_{w_J}\cap \mathrm{Pet}_n,
\]
with
\[
X_J\cap Q_K\neq \varnothing \iff J\supset K,
\qquad
X_J\cap Q_J=\{w_J\}.
\]
The corresponding cohomology classes
\[
w_J := \frac{1}{m_J}\prod_{i\in J} w_i
\]
form a \(\mathbb Z\)-basis of \(H^*(\mathrm{Pet}_n;\mathbb Z)\), are dual to \([X_J]\), and coincide with the Harada–Tymoczko basis:
\[
p_{v_J}=w_J.
\]
The structure constants satisfy
\[
d_{JK}^L\ge 0,
\]
and admit a manifestly positive formula
\[
d_{JK}^L = \frac{m_L}{m_Jm_K} \sum_{P\in \mathcal A_{JK}^L}\mathrm{wt}(P)
\]
in terms of left-right diagrams [2104.02914].

## 6. Extensions: mixed \(\Phi\)-Eulerian numbers and the broader Peterson program

Peterson Schubert calculus also computes mixed \(\Phi\)-Eulerian numbers. For a root system \(\Phi\), the mixed numbers are the coefficients in the volume polynomial
\[
V_\Phi(u_1,\dots,u_n) = \sum_{c_1+\cdots+c_n=n} A^\Phi_{c_1,\dots,c_n}\, \frac{u_1^{c_1}}{c_1!}\cdots \frac{u_n^{c_n}}{c_n!}.
\]
They can be realized as Peterson intersections:
\[
A^\Phi_{c_1,\dots,c_n} = \int_{\mathrm{Pet}_\Phi} w_1^{c_1}\cdots w_n^{c_n},
\]
and
\[
\int_{\mathrm{Pet}_\Phi} w_1\cdots w_n=\det(C_\Phi).
\]
In type \(A\), this reproduces the Berget–Spink–Tseng computation, while in arbitrary Lie type the recursive computation is governed by Peterson Monk/Giambelli rules [2104.14083].

The type-uniform matrix formula for Peterson structure constants immediately yields a type-uniform formula for mixed \(\Phi\)-Eulerian numbers:
\[
A_{c_1,\dots,c_n}^\Phi = \frac{|W_\Phi|}{\det(C_\Phi)} \left[ \left((m_{1,J}^K)_{J,K}\right)^{c_1} \cdots \left((m_{n,J}^K)_{J,K}\right)^{c_n} \right]_{(\emptyset,\Delta)}.
\]
This ties weight-polytope volumes directly to Peterson multiplication [2508.05457].

The subject also sits inside a broader Peterson program. One branch identifies the spectrum of \(QH^*(G/P)\) 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
\[
X=X_{w_0s_{n-1}}\subset F\ell_n,
\]
the paper [2509.17857] gives a Borel-type presentation of \(QH^*(X)\), a quantum Chevalley formula, and the statement that the quantum Schubert polynomials for \(X\) are the same as those for \(F\ell_n\). The paper explicitly says that its presentation was predicted in the extended Peterson perspective of Li–Rietsch–Yang [2509.17857].

A related affine direction is the Peterson–Lam–Shimozono theorem, interpreted as an affine analogue of the quantum Chevalley formula. It identifies \(T\)-equivariant quantum Schubert calculus on \(G/P\) with \(T\)-equivariant affine Schubert calculus on the affine Grassmannian \(\mathcal G r\), via a map sending
\[
\sigma_{w t_\lambda}\mapsto
\begin{cases}
q_\lambda\,\sigma^w & \text{if } w t_\lambda\in (W^P)_{\mathrm{af}},\\
0 & \text{otherwise}.
\end{cases}
\]
This places Peterson Schubert calculus within a larger network linking flag varieties, affine Schubert geometry, and quantum multiplication [2110.09985].

Taken together, these developments present Peterson Schubert calculus as a subvariety analogue of classical Schubert calculus on \(G/B\): 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 [1309.0484], [1311.3014], [2106.10372], [2508.05457].

Source: https://www.emergentmind.com/topics/peterson-schubert-calculus