2.1. The integration pairing

Each irreducible, T-stable, closed subvariety $Z \subset X$ of complex dimension k has a fundamental class $[Z]_T \in H_{2k}^T(X)$ . If X is smooth and irreducible, then there exists a unique class $\eta _Z \in H^{2 (\dim X - k)}_T(X)$ , called the Poincaré dual of Z, such that

$$ \begin{align*}\eta_Z \cap [X]_T = [Z]_T.\end{align*} $$

Given a T-equivariant proper map $f:X\to Y$ , there is a push-forward $f_*:H_i^T(X) \to H_i^T(Y)$ , determined by the fact that if $Z \subset X$ is irreducible and T-stable, then

$$ \begin{align*} f_*([Z])= \begin{cases} d_Z[f(Z)]&\text{if }\dim f(Z)=\dim Z,\\ 0&\text{if }\dim f(Z)<\dim Z, \end{cases} \end{align*} $$

where $d_Z$ is the generic degree of the restriction $f:Z \to f(Z)$ . The push-forward and pull-back are related by the projection formula

(7) $$ \begin{align} f_* (f^*(\eta) \cap c) = \eta \cap f_*(c), \end{align} $$

for $\eta \in H^*_T(Y)$ and $c\in H^T_*(X)$ . Recall that we have an isomorphism

(8) $$ \begin{align} &&H^j_T(pt)\xrightarrow\sim H_{-j}^T(pt),&& a\mapsto a\cap[pt]_T. \end{align} $$

In particular, $H^*_T(pt)$ lives in non-negative degrees, and $H_*^T(pt)$ lives in non-positive degrees.

Suppose now that X is complete, so that $f: X \to pt$ is proper. For a homology class $c \in H_{-j}^T(X)$ , we denote by $\int _X c$ the class $f_*(c) \in H_{-j}^T(pt)$ , viewed as an element of $H_T^{j}(pt)$ via Equation (8). Then, we may define a pairing,

(9) $$ \begin{align} \langle\ , \,\rangle : H^i_T(X) \mathop \times H_{-j}^T(X) \to H^{i+j}_T(pt); \quad \langle \eta, c \rangle := \int_X \eta \cap c. \end{align} $$

The pairing in Equation (9) is compatible with the pairing in ordinary (co)homology. We have forgetful maps $H^*_T(X)\to H^*(X)$ and $H_*^T(X)\to H_*(X)$ , and a commutative diagram,

2.2. Spaces with affine paving

Following [Reference Fulton20, Example 1.9.1] (see also [Reference Graham23]), we say that a T-variety X admits a T-stable affine paving if it admits a filtration $X:=X_n \supset X_{n-1} \supset \dots $ by closed T-stable subvarieties such that each is a finite disjoint union of T-invariant varieties $U_{ij}$ isomorphic to affine spaces $\mathbb A^i$ .

2.3. Chern classes and Euler classes

We will denote by $c_i^T(\ )$ the ith T-equivariant Chern class, and by $e^T(\ )$ the equivariant Euler class of a T-equivariant vector bundle. We say that a section s of a vector bundle $\mathcal V\to X$ is regular if the codimension of the zero set of s equals the rank of $\mathcal V$ . Recall that the torus T acts on the space $H^0(X,\mathcal V)$ of sections of $\mathcal V$ via the formula $(z\cdot s)(x) = zs(z^{-1}x)$ for all $x\in X$ and $z\in T$ . The sections which are invariant under this action are precisely those that intertwine the action, that is, satisfy $s(zx)=zs(x)$ for all $x\in X$ and $z\in T$ .

Lemma 2.2 (cf. [Reference Graham and Kreiman24, Lemma 2.2])

If $\mathcal L$ is a T-equivariant line bundle on a T-scheme X, and s is a T-invariant regular section of $\mathcal L$ with zero scheme Y, then

$$ \begin{align*} [Y]_T=c_1^T(\mathcal L)\cap[X]_T. \end{align*} $$

For $\lambda $ a character of T, let $\underline {\mathbb C}_{\lambda } = X\times \mathbb C\to X$ denote the (geometrically trivial) equivariant line bundle, with T-action given by $z(x,v)=(zx,\lambda (z)v)$ for all $z\in T$ . By a standard abuse of notation, we write $\lambda $ for the T-equivariant first Chern class of $\underline {\mathbb C}_{\lambda }$ .

Corollary 2.4. Let $\mathcal V\to X$ be a T-equivariant vector bundle. For a character $\lambda $ of T, let s be a regular section of $\mathcal V$ that lies in the $\lambda $ -weight space, that is, $(z\cdot s)=\lambda (z) s $ for all $z\in T$ . The zero scheme $Z(s)$ of s is T-invariant, and we have

$$ \begin{align*} [Z(s)]_T=e^{T}\left(\mathcal V\otimes\underline{\mathbb C}_{-\lambda}\right)\cap[X]_T. \end{align*} $$

If $\mathcal V$ admits a filtration with T-equivariant line bundle quotients $\{\mathcal L_i\}$ , we have

$$ \begin{align*} [Z(s)]_T=\prod\left(c_{1}^{T}(\mathcal L_i)-\lambda\right)\cap[X]_T. \end{align*} $$

Proof. If $\lambda $ is non-trivial, the section s is not invariant. Observe however that the section $r=s\otimes 1$ of the vector bundle $\mathcal V\otimes \underline {\mathbb {C}}_{-\lambda }$ is T-invariant, since

$$ \begin{align*} r(z x)&=s(z x)\otimes 1=\left(zz^{-1}s(zx)\right)\otimes 1\\ &= \left(z\left( \left(z^{-1}\cdot s\right)(x) \right)\right) \otimes 1 \\ &= \left(z\left( \lambda(z^{-1})s(x) \right)\right)\otimes 1\\ & = \left(\lambda(z^{-1})zs(x)\right)\otimes 1 \\ &=zs(x)\otimes \lambda(z^{-1})=z(s(x)\otimes 1)=zr(x). \end{align*} $$

Further, r has the same zero scheme as s, that is, $Z(r)=Z(s)$ . Hence, the first equality follows from Corollary 2.3.

Suppose $\mathcal V$ admits a filtration by line bundles $\{\mathcal L_i\}$ . Then, $\mathcal V\otimes \underline {\mathbb {C}}_{-\lambda }$ admits a filtration by line bundles $\{\mathcal L_i\otimes \underline {\mathbb {C}}_{-\lambda }\}$ . Applying the Whitney splitting principle, we have

$$ \begin{align*} e^T\left(\mathcal V\otimes\underline{\mathbb{C}}_{-\lambda}\right) =\prod\left(c_1^{T}\left(\mathcal L_i\otimes\underline{\mathbb{C}}_{-\lambda}\right)\right) =\prod\left(c_1^{T}(\mathcal L_i)-\lambda\right), \end{align*} $$

from which the second equality follows.

3.1. Flag manifolds and Schubert varieties

The flag manifold $G/B$ is a projective algebraic manifold with a transitive action of G given by left multiplication. It has a stratification into finitely many B-orbits (resp. $B^-$ -orbits) called the Schubert cells $X_w^\circ := BwB/B$ (resp. $X^{w,\circ }:= B^- wB/B $ ), that is,

(10)

The closures $X_w:=\overline {X_w^\circ }$ and $X^w:=\overline {X^{w,\circ }}$ are called Schubert varieties. The Bruhat order is a partial order on W characterized by inclusions of Schubert varieties, that is, $X_v \subset X_w$ if and only if $v \le w$ , and $X^w\subset X^v$ if and only if $v\leq w$ . Following Lemma 2.1, the fundamental classes (resp. ) form a basis of $H_*^T(X_w)$ (resp. $H_*^T(X^w)$ ).

The cohomology classes $\sigma _v^T\in H_T^*(X)$ Poincaré dual to the $[X^v]_T$ , that is, characterized by the equation $\sigma _v^T\cap [G/B]_T=[X^v]_T$ , are called Schubert classes. Following Lemma 2.1, the Schubert classes form a basis of $H_T^*(G/B)$ as a module over $H_T^*(pt)$ .

3.2. Line bundles on the flag manifold

Recall that since G is simply connected, the character group $\mathfrak X(T)$ of T equals the weight lattice of $\Delta $ . For $\lambda \in \mathfrak X(T)$ , let $\mathbb C_\lambda $ be the one-dimensional B-representation on which T acts via the character $\lambda $ , and the unipotent radical of B acts trivially. We will denote by $\mathcal L_\lambda $ the T-equivariant line bundle

$$ \begin{align*} \mathcal L_\lambda:=G\times^B\mathbb C_{-\lambda}\to G/B,&&(g,v)\mapsto gB, \end{align*} $$

with T-action given by $t\cdot (g,v)=(tg,v)$ .

3.3. Stability of Dynkin diagrams

Let $(\,|\,)$ be a positive-definite W-invariant bilinear form on ${\mathfrak {g}}$ . For $\alpha \in \Delta $ , the corresponding coroot is given by $\alpha ^\vee =\frac {2(\alpha |\,\_)}{(\alpha |\alpha )}$ . Recall that the Cartan matrix is a square matrix with rows indexed by simple roots, with the $\alpha \beta $ th entry given by $a_{\alpha \beta }=\left \langle \beta ^\vee ,\alpha \right \rangle =\frac {2(\alpha |\beta )}{(\alpha |\alpha )}$ .

For $I\subset \Delta $ , the Cartan matrix of I is the submatrix of $\Delta $ spanned by the rows and columns indexed by the roots in I. Let $\Phi ^\vee $ denote the set of coroots, and let $\Phi ^\vee _I$ denote the subset of coroots corresponding on I. The pairing $\langle \ ,\,\rangle $ on $\Phi _I^\vee \times \Phi _I$ is the restriction of the pairing $\Phi ^\vee \times \Phi $ to $\Phi _I\subset \Phi $ and $\Phi ^\vee _I\subset \Phi ^\vee $ . We describe this by saying that the roots and coroots are stable for the inclusion of Dynkin diagrams.

Consider the elements $\varpi ^I_\alpha \in \bigoplus \limits _{\alpha \in I}\mathbb Q\alpha $ and $\varpi ^{I\vee }_\alpha \in \bigoplus \limits _{\alpha \in I}\mathbb Q\alpha ^\vee $ given by the equations

$$ \begin{align*} &&\langle\varpi^{I\vee}_\alpha,\beta\rangle=\langle\beta^\vee,\varpi^I_\alpha\rangle =\delta_{\alpha\beta}&&\forall\beta\in I. \end{align*} $$

Then, $\varpi _\alpha :=\varpi _\alpha ^\Delta $ is the fundamental weight dual to $\alpha ^\vee $ , and $\varpi _\alpha ^\vee :=\varpi _\alpha ^{\Delta \vee }$ is the fundamental coweight dual to the root $\alpha $ . In general,

$$ \begin{align*} &&\varpi_\alpha^I\neq\varpi_\alpha&& \text{and}&& \varpi^{I\vee}_\alpha\neq\varpi^\vee_\alpha.&& \end{align*} $$

We express this fact by saying that the fundamental weights and coweights are not stable for the inclusion of Dynkin diagrams.

3.4. The height function

Let

$$ \begin{align*} \rho_I=\frac 12\sum_{\alpha\in\Phi_I^+}\alpha=\sum_{\alpha\in I}\varpi_\alpha^I, && \rho_I^\vee=\frac 12\sum_{\alpha\in\Phi_I^+}\alpha^\vee=\sum_{\alpha\in I}\varpi_\alpha^{I\vee}. \end{align*} $$

We set $\rho =\rho _\Delta $ and $\rho ^\vee =\rho ^\vee _\Delta $ . Following [Reference Bourbaki11, Chapter 6, Proposition 29], we have $\langle \rho ^\vee ,\alpha \rangle =1$ for $\alpha \in \Delta $ . For $\lambda =\sum _{\alpha \in \Delta }a_\alpha \alpha $ , we define the height of $\lambda $ to be

$$ \begin{align*} ht(\lambda)=\sum a_\alpha=\langle\rho^\vee,\lambda\rangle. \end{align*} $$

Let $h=2\rho ^\vee $ , and let $\mathfrak s\subset {\mathfrak {g}}$ be the Lie subalgebra spanned by h. Observe that h is in the coroot lattice, and hence there exists a one-dimensional sub-torus $S\subset T$ with $Lie(S)=\mathfrak s$ . For any $\alpha ,\beta \in \Delta $ , we have $\left \langle h,\alpha \right \rangle =\left \langle h,\beta \right \rangle $ , and hence $\alpha |\mathfrak s=\beta |\mathfrak s$ . Let $t=\alpha |\mathfrak s$ for some $\alpha \in \Delta $ . The restriction map $\mathfrak h^*\to \mathfrak s^*$ (dual to the inclusion $\mathfrak s\hookrightarrow \mathfrak h$ ) satisfies $\alpha \mapsto t$ for all $\alpha \in \Delta $ , and hence is given by $\lambda \mapsto ht(\lambda )t=\left \langle \rho ^\vee ,\lambda \right \rangle t$ .

4.1. Hessenberg varieties

Let ${\mathfrak {g}}:= Lie(G)$ , $\mathfrak b=Lie(B)$ , and $\mathfrak h:=Lie(T)$ . A subspace $H\subset {\mathfrak {g}}$ is called a Hessenberg space if it is B-stable and if $\mathfrak b\subset H$ . Let

$$ \begin{align*} H_0=\mathfrak b\oplus\bigoplus\limits_{\alpha\in\Delta}{\mathfrak{g}}_{-\alpha}. \end{align*} $$

We say that a Hessenberg space H is indecomposable if $H_0\subset H$ . Recall that the vector bundle $G\times ^B{\mathfrak {g}}\to G/B$ is trivialized by the map

$$ \begin{align*} \mu_{\mathfrak{g}}:G\times^B{\mathfrak{g}}\to{\mathfrak{g}},&&(g,x)\mapsto Ad(g)x, \end{align*} $$

that is, we have an isomorphism $G\times ^B{\mathfrak {g}}\to G/B\times {\mathfrak {g}}$ given by $(g,x)\mapsto (gB,Ad(g)x)$ . Let H be a Hessenberg space, and let $\mu _H$ denote the restriction of $\mu _{{\mathfrak {g}}}$ to the sub-bundle $G\times ^BH\subset G\times ^B{\mathfrak {g}}$ .

For $x\in {\mathfrak {g}}$ , the fiber $\mu _H^{-1}(x)$ (viewed as a subscheme of $G/B$ ) is called the Hessenberg scheme $\mathbf H(x,H)$ . If H is indecomposable, $\mathbf H(x,H)$ is reduced and irreducible for all x (see [Reference Abe, Fujita and Zeng1, Theorem 1.2]). In this case, we call $\mathbf H(x,H)$ a Hessenberg variety.

For the rest of this article, we assume without mention that the Hessenberg space H is indecomposable.

For each positive simple root $\alpha \in \Delta $ , choose a root vector $e_\alpha \in \mathfrak {g}_\alpha $ . Set

(11) $$ \begin{align} &e= \sum_{\alpha \in\Delta} e_\alpha,& h=2\rho^\vee.& \end{align} $$

The element e is a regular nilpotent element in $\mathfrak b$ (see, e.g., [Reference Collingwood and McGovern14], [Reference Kostant32]). Recall from Section 3.4 the sub-torus $S\subset T$ lifting the element $h\in \mathfrak h$ . Following [Reference Bourbaki11, Chapter 6, Proposition 29], we have $\langle \rho ^\vee ,\alpha \rangle =1$ , and hence

$$ \begin{align*} [h,e]=[h,\sum_{\alpha\in\Delta} e_\alpha]=\sum_{\alpha\in\Delta} \langle h,\alpha\rangle e_\alpha=2e. \end{align*} $$

We see that the vector space $\mathbb C e$ is h-stable, and hence also S-stable. Consequently, the Hessenberg variety $\mathbf H(e,H)$ is S-stable. Since $h\in \mathfrak h$ , the adjoint action of T on $\mathfrak s=\mathbb C h$ is trivial. In particular, $\mathfrak s$ is T-stable, and hence so is $\mathbf H(h,H)$ . The following result was proved in type A by Anderson and Tymoczko in [Reference Anderson and Tymoczko6].

Theorem 4.1. For any indecomposable Hessenberg space H, we have

$$ \begin{align*} [\mathbf H(h,H)]_T&=\prod\left(c_1^T(\mathcal L_\alpha)\right)\cap[G/B]_T,\\ [\mathbf H(e,H)]_S&=\prod\left(c_1^S(\mathcal L_\alpha)-t\right)\cap[G/B]_S, \end{align*} $$

where the product is over the set .

4.2. The Peterson variety

The Peterson variety is defined by

(12) $$ \begin{align} \mathbf{P}:= \mathbf H(e,H_0)\subset G/B. \end{align} $$

It is a subvariety of G/B of dimension $rk(G)=|\Delta |$ , singular in general. Following Theorem 4.1, the S-equivariant fundamental class of the Peterson variety in $H_*^S(G/B)$ is given by

(13) $$ \begin{align} [\mathbf{P}]_S=\prod\limits_{\alpha\in\Phi^+\backslash\Delta}\left(c_1^S\left(\mathcal L_\alpha\right)-t\right)\cap[G/B]_S. \end{align} $$

Let $\mathbf {P}_I$ denote the Peterson variety corresponding to the Dynkin diagram $I\subset \Delta $ . The following was proved in classical types by Tymoczko [Reference Tymoczko41, Theorem 4.3] and generalized to all Lie types by Precup [Reference Precup38] (see also [Reference Goldin, Mihalcea and Singh22, Appendix A]).

The subvarieties $\mathbf {P}^\circ _I$ (called Peterson cells) are S-stable affine spaces. The Peterson cell $\mathbf {P}^\circ _I$ has a unique S-stable point $w_I$ , the longest element in the Weyl subgroup $W_I$ . Following Lemma 2.1, the fundamental classes form a basis of $H_*^S(\mathbf {P})$ over $H^*_S(pt)$ .

Proposition 4.3 [Reference Goldin, Mihalcea and Singh22, Theorem 4.3]

Consider the inclusion $i:\mathbf {P}\hookrightarrow G/B$ . For each $I\subset \Delta $ , fix a Coxeter element $v_I$ for I. There exist positive integers $m(v_I)$ such that

$$ \begin{align*} \left\langle i^*\sigma_{v_I}^S,[\mathbf{P}_J]_S\right\rangle=m(v_I)\delta_{IJ}. \end{align*} $$

In particular, is a basis for $H^*_S(\mathbf {P})$ . Furthermore, the numbers $m(v_I)$ do not depend on the superset $\Delta $ containing I.

Proposition 4.4 deals with the stability of Schubert classes and their pullbacks to the Peterson variety. For $I\subset \Delta $ , let $G_I\subset G$ be the standard Levi subgroup, and $B_I:=B\cap G_I$ the corresponding Borel subgroup of $G_I$ .

For $w\in W_I$ , Proposition 4.4 allows us to abuse notation and denote by $p_w$ both the class $i^*\sigma _w^S$ in $H_S^*(\mathbf {P})$ and its pullback $p_w^I$ in $H^*_S(\mathbf {P}_I)$ .

5.1. Cohomology of regular Hessenberg varieties

The goal of Section 5.1 is primarily expository. Let H be any (indecomposable) Hessenberg space, and let h and e be as in Equation (11). Klyachko [Reference Klyachko30], [Reference Klyachko31] and Tymoczko [Reference Tymoczko42] have constructed an action of W on $H^*(\mathbf H(h,H))$ , called the Tymoczko dot-action. Abe, Horiguchi, Masuda, Murai, and Sato [Reference Abe, Horiguchi, Masuda, Murai and Sato4] proved a commutative diagram relating the cohomologies of $G/B$ and certain Hessenberg varieties (see Proposition 5.3). This relationship between the cohomologies of the regular nilpotent and regular semisimple Hessenberg varieties was earlier described in type A by Abe, Harada, Horiguchi, and Masuda in [Reference Abe, Harada, Horiguchi and Masuda2]. A proof of Proposition 5.3 for s regular, semisimple in a Euclidean neighborhood of e can be found in the works of Brosnan and Chow [Reference Brosnan and Chow12] and Bălibanu and Crooks [Reference Bălibanu and Crooks8]. We show that the cohomology of the regular nilpotent Hessenberg variety is isomorphic to the Weyl-invariant part of the cohomology of the regular semisimple Hessenberg variety, $\mathbf H(h, H)$ .

Recall the map $\mu _H:G\times ^BH\to {\mathfrak {g}}$ given by $(g,x)\mapsto Ad(g)x$ . Let ${\mathfrak {g}}^r$ denote the set of regular elements in ${\mathfrak {g}}$ , and let $H^r=H\cap {\mathfrak {g}}^r$ . Following [Reference Bălibanu and Crooks8, Section 4], for any $x\in {\mathfrak {g}}$ , there exists a Euclidean open neighborhood $D_x$ of x, such that the (non-equivariant) inclusion $\mu _H^{-1}(x)\hookrightarrow \mu _H^{-1}(D_x)$ induces an isomorphism

(14) $$ \begin{align} H^*(\mu_H^{-1}(D_x))\overset\sim\longrightarrow H^*(\mu_H^{-1}(x)). \end{align} $$

Let $Z=\mu _H^{-1}(D_x)\cap (G\times ^BH^r)$ , let s be a regular semisimple element contained in $D_x$ , and consider the commutative diagram

(15)

Composing the induced pullback map $H^*(\mu _H^{-1}(D_x))\to H^*(\mathbf H(s,H))$ with the isomorphism (14), we obtain the so-called local invariant cycle map

$$ \begin{align*} \lambda_x:H^*(\mathbf H(x,H))\to H^*(\mathbf H(s,H))^W. \end{align*} $$

Following [Reference Bălibanu and Crooks8], an application of the local invariant cycle theorem of Beilinson, Bernstein, and Deligne [Reference Beĭlinson, Bernstein and Deligne9] yields the following result.

Proposition 5.3. We have a commutative diagram

(16)

Proof. For $x=e$ , there exists a regular semisimple element s in a neighborhood of e such that the map $\lambda _e:H^*(\mathbf H(e,H))\to H^*(\mathbf H(s,H))^W$ is an isomorphism (cf. [Reference Bălibanu and Crooks8, Proposition 4.7]). Observe that $\mathbf H(e,H)\hookrightarrow Z$ in this case. Thus, we have the following commutative diagram:

Following Diagram (15), the pullbacks $i^*:H^*(G/B)\to H^*(\mathbf H(e,H))$ and $j^*:H^*(G/B)\to H^*(\mathbf H(s,H))$ factor through the pullback $H^*(G/B)\to H^*(Z)$ , and hence we have a commutative diagram

(17)

We conjugate s to a regular semisimple element $s'\in T$ . Note that the corresponding Hessenberg varieties $\mathbf H(s, H)$ and $\mathbf H(s',H)$ are isomorphic varieties, but they have conjugate torus actions. Since $s'$ and h are both regular and semisimple in T, we invoke the isomorphism described in [Reference Abe, Horiguchi, Masuda, Murai and Sato4] between $H_T^*(\mathbf H(s',H))$ and $H^*_T(\mathbf H(h,H))$ that descends to an isomorphism on the ordinary cohomology:

Here, the vertical maps are each the surjective maps naturally induced by forgetting the T action. The bottom horizontal isomorphism is invariant with respect to the W-action on the cohomology, inducing an isomorphism between $H^*(\mathbf H(s,H))^W$ and $H^*(\mathbf H(h,H))^W$ . The commuting diagram (16) then follows by the commuting diagram (17) together with these isomorphisms.

5.2. Giambelli and multiplicity formulas

We now specialize to the case $\mathbf H(e,H)=\mathbf {P}$ and $\mathbf H(h,H)=\mathbf {Perm}$ . There are two parts to the proof of the equivariant Giambelli formula, namely, showing that there is a single non-zero coefficient, and computing this coefficient. The calculation of the non-zero coefficient is deduced from the analogous result in [Reference Klyachko30] for the ordinary cohomology of the permutahedral variety. The claim that there is a single non-zero coefficient is essentially from [Reference Drellich17], where the result is proved only for specific choices of $v_K$ , but the proofs follow almost verbatim.

Lemma 5.4 (Ordinary Giambelli formula)

Let $v_I$ be a Coxeter element for $I\subset \Delta $ , and let $R(v_I)$ be the number of reduced words for $v_I$ . We have

(18) $$ \begin{align} i^*\sigma_{v_I}=\frac{R(v_I)}{|I|!}\prod_{\alpha\in I}i^*\sigma_\alpha. \end{align} $$

Proof. Let $\mathcal R(v_I)$ denote the set of reduced words for $v_I$ . Following [Reference Klyachko30], [Reference Klyachko31] (see also [Reference Nadeau and Tewari35, Theorem 8.1]), we have

$$ \begin{align*} j^*\sigma_{v_I}=\frac{1}{\ell(v_I)!}\sum_{\underline v\in\mathcal R(v_I)}\prod_{s_\alpha\in\underline v}j^*\sigma_\alpha, \end{align*} $$

where $\ell (\_)$ denotes the length function on W. Since $v_I$ is a Coxeter word for I, we have $\ell (v_I)=|I|$ . Further, every reduced word $\underline v\in \mathcal R(v_I)$ contains each simple reflection exactly once. Hence, we have

(19) $$ \begin{align} \frac{1}{\ell(v_I)!}\sum_{\underline v\in\mathcal R(v_I)}\prod_{s_\alpha\in\underline v}j^*\sigma_\alpha =\frac{R(v_I)}{|I|!}\prod_{\alpha\in I}j^*\sigma_\alpha. \end{align} $$

Finally, it follows from Equation (16) that the pullbacks $i^*:H^*(G/B)\to H^*(\mathbf {P})$ and $j^*:H^*(G/B)\to H^*(\mathbf {Perm})$ have the same kernel, and hence any relation amongst the classes $j^*\sigma _w$ also holds amongst the classes $i^*\sigma _w$ . The claim now follows from Equation (19).

Recall that we denote by $p_w$ the pullback class $i^*\sigma _w^S$ , and by $p_\alpha $ the pullback class $i^*\sigma _{\alpha }^S$ . For convenience, we write the equivariant class

$$ \begin{align*} \Omega_I:=\prod_{\alpha\in I}p_\alpha = \prod_{\alpha\in I}i^*(\sigma^S_{\alpha}) \end{align*} $$

for any $I\subset \Delta $ , where $i^*: H_S^*(G/B)\rightarrow H^*_S(\mathbf {P})$ is the pullback map in equivariant cohomology induced by the inclusion $i:\mathbf {P}\to G/B$ .

Theorem 5.5 (Equivariant Giambelli formula)

Let $v_I$ be a Coxeter element for $I\subset \Delta $ , and let $R(v_I)$ be the number of reduced words for $v_I$ . We have

$$ \begin{align*} p_{v_I}=\frac{R(v_I)}{|I|!} \Omega_I. \end{align*} $$

Proof. Observe that the restriction to ordinary cohomology $H^*_S(\mathbf {P})\to H^*(\mathbf {P})$ is given by

$$ \begin{align*} \Omega_J\mapsto \prod_{\alpha\in J}i^*\sigma_\alpha. \end{align*} $$

Consider $J\subseteq \Delta $ . Following Lemma 5.4 and Proposition 4.3,

is a basis of $H^*(\mathbf {P}_J)$ . Since $H^*_S(\mathbf {P}_J)$ (Lemma 2.1) is a free module over $\mathbb Q[t]$ , by equivariant formality, is a basis of $H^*_S(\mathbf {P}_J)$ over $\mathbb Q[t]$ .

Consider now the basis expansion

(20) $$ \begin{align} p_{v_I}=i^*\sigma_{v_I }^{S}=\sum_{J\subseteq\Delta} c_{v_I }^J\Omega_J \end{align} $$

in $H^*_S(\mathbf {P})$ , where the $c_{v_I}^J\in \mathbb Q[t]$ are t-monomials of degree $|I|-|J|$ .

First, consider J for which $I\subsetneq J$ . Then, $|I|-|J|<0$ , and hence $c_{v_I}^J= 0$ .

Next, consider the case $J=I$ . The coefficient $c_{v_I}^I$ has degree $0$ , hence can be obtained from the specialization $H^*_S(\mathbf {P})\to H^*(\mathbf {P})$ . Applying Lemma 5.4,

$$ \begin{align*} c_{v_I }^I=\frac{R(v_I)}{|I|!}. \end{align*} $$

Finally, suppose J satisfies $I\not \subseteq J$ . Consider the localization of $\sigma _{v_I}$ to the fixed point w. Then, $\sigma _{v_I}|_w\neq 0$ implies $v_I\leq w$ in the Bruhat order (see [Reference Billey10]). In particular, for all fixed points $w\in \mathbf {P}_J$ , we have $\sigma _{v_I}|_w=0$ . By injectivity of localization (see [Reference Drellich17, Theorem 3.1]), the pull-back of $\sigma _{v_I}$ to $\mathbf {P}_J$ is zero. Recall that the pull-back $H^*_S(\mathbf {P})\to H^*_S(\mathbf {P}_J)$ sends $\Omega _K$ to $\Omega _K$ if $K\subset J$ , and to $0$ otherwise. Applying the pull-back, Equation (20) yields

$$ \begin{align*}0 = \sum_{K\subseteq J} c_{v_I }^K \Omega_K \end{align*} $$

in $H^*_S(\mathbf {P}_J)$ . Since the set forms a basis of $H^*_S(\mathbf {P}_J)$ , we deduce that $c_{v_I}^K=0$ for all $K\subset J$ ; in particular, $c_{v_I}^J=0$ .

Proof. Observe that since $\deg \left ([\mathbf {P}]_S\right )=\deg \left (\Omega _\Delta \right )=|\Delta |$ , we have $ \left \langle \Omega _\Delta ,[\mathbf {P}]_S\right \rangle = \left \langle \prod i^*\sigma _{\alpha },[\mathbf {P}]\right \rangle , $ where the latter expression is the pairing in ordinary (co)homology. Observe that $[\mathbf {P}]=[\mathbf {Perm}]$ in $H_*(G/B)$ by [Reference Abe, Fujita and Zeng1]. (Alternatively, apply the forgetful maps $H^*_T(G/B)\to H^*(G/B)$ and $H^*_S(G/B)\to H^*(G/B)$ to the equations in Theorem 4.1 to draw the same conclusion.) It follows that

$$ \begin{align*} \left\langle\Omega_\Delta,[\mathbf{P}]_S\right\rangle= \left\langle\prod_{\alpha\in\Delta} i^*\sigma_\alpha,[\mathbf{P}]\right\rangle= \left\langle\prod_{\alpha\in\Delta} i^*\sigma_\alpha,[\mathbf{Perm}]\right\rangle=\frac{|W|}{f_\Delta}, \end{align*} $$

where the latter equality is from [Reference Klyachko30, Theorem 3].

Theorem 5.7 (Multiplicity formula)

For $v_I$ a Coxeter element of I, we have

$$ \begin{align*} \left\langle p_{v_I},[\mathbf{P}_I]_S\right\rangle=\frac{R(v_I)|W_I|}{|I|!\,f_I}. \end{align*} $$

Proof. Since $m(v_I)$ does not depend on the diagram $\Delta $ containing I, we may assume $I=\Delta $ . Using Theorem 5.5 and Lemma 5.6, we obtain

$$ \begin{align*} \left\langle p_{v_I},[\mathbf{P}_I]_S\right\rangle =\frac{R(v_I)}{|I |!}\left\langle\Omega_I ,[\mathbf{P}_I]\right\rangle=\frac{R(v_I)|W_I|}{|I |!f_I}. \end{align*} $$

6.1. Dual Peterson classes

In Theorem 6.3, we compute the equivariant cohomology class in $H^*_S(\mathbf {P})$ which is dual to the Peterson subvariety $\mathbf {P}_I$ . This is a key step in our proof of the Chevalley formula.

Proof. Let $\varpi _\alpha $ be the fundamental weight dual to the coroot $\alpha ^\vee $ , let $V_{\varpi _\alpha }$ be the corresponding irreducible G-representation, and let $\operatorname {\mathrm {pr}}:V_{\varpi _\alpha }\to \mathbb C_{-\varpi _\alpha }$ be the B-equivariant projection onto the lowest weight space in $V_{\varpi _\alpha }$ .

Let $\mathbf 1\in \mathbb C_{-\varpi _\alpha }$ be a lowest weight vector in $V_{\varpi _\alpha }$ , and consider the section s of the line bundle $\mathcal L_{\varpi _\alpha }\to G/B$ given by $s(gB)= (g,\operatorname {\mathrm {pr}}(g^{-1}\mathbf 1))$ . Observe that the torus T acts on s via the character $-\varpi _\alpha $ :

$$ \begin{align*} (z\cdot s)(gB) & =zs(z^{-1}gB) = z(z^{-1}g,\operatorname{\mathrm{pr}}(g^{-1}z\mathbf 1))\\ & =z(z^{-1}g,\varpi_\alpha(z^{-1})\operatorname{\mathrm{pr}}(g^{-1}\mathbf 1))\\ & =\varpi_\alpha(z^{-1}) (g,\operatorname{\mathrm{pr}}(g^{-1}\mathbf 1)) =\varpi_\alpha(z^{-1})s(gB). \end{align*} $$

Further, the zero scheme $Z(s)$ of s is supported precisely on the Schubert divisor $X^{s_\alpha }$ , thus $[Z(s)]_T=m[X^{s_\alpha }]_T$ for some positive integer m. It follows from Corollary 2.4 that

(21) $$ \begin{align} m\sigma^T_\alpha=c_1^T(\mathcal L_{\varpi_\alpha})+\varpi_\alpha. \end{align} $$

We evaluate Equation (21) under localization $\ell _{s_\alpha }^*:H^*_T(G/B)\to H^*(\{s_\alpha \})$ at the T-fixed point $s_\alpha $ . Recall that $\mathcal L_{\varpi _\alpha }=G\times ^B\mathbb C_{-\varpi _\alpha }$ , and hence $\ell _{s_\alpha }^*(c_1(\mathcal L_{\varpi _\alpha }))=s_\alpha (-\varpi _\alpha )=-\varpi _\alpha +\alpha $ . Using the localization formula of Andersen, Jantzen, and Soergel [Reference Andersen, Jantzen and Soergel5], and Billey [Reference Billey10], we have $\ell _{s_\alpha }^*(\sigma _\alpha )=\alpha $ . It follows that $m=1$ , that is,

$$ \begin{align*} c_1^T(\mathcal L_{\varpi_\alpha})=\sigma^T_\alpha-\varpi_\alpha. \end{align*} $$

Recall that $\alpha =\sum a_{\alpha \beta }\varpi _\beta $ , where $a_{\alpha \beta }=\left \langle \beta ^\vee ,\alpha \right \rangle $ is the $\alpha \beta $ th entry of the Cartan matrix. We have

$$ \begin{align*} c_1^T(\mathcal L_\alpha) =\sum a_{\alpha\beta} c_1^T(\mathcal L_{\varpi_\beta}) =\sum a_{\alpha\beta}\sigma^T_\beta-\sum a_{\alpha\beta}\varpi_\beta =\sum a_{\alpha\beta}\sigma^T_\beta-\alpha. \end{align*} $$

Consequently, we have $ c_1^S(\mathcal L_\alpha ) =\sum a_{\alpha \beta }\sigma ^S_\beta -t. $ Applying the S-equivariant pullback $i^*$ , we obtain the claimed equality, $c_1^S(i^*\mathcal L_\alpha )=q_\alpha -t$ .

Proposition 6.2. For $I\subset \Delta $ , we have the following equality in $H_*^S(G/B)$ :

$$ \begin{align*} \prod\limits_{\alpha\in\Phi^+\backslash\Phi_I^+}\left(c_1^S(\mathcal L_\alpha)-t\right)\cap[G/B]_S= \frac{|W|}{|W_I|}[X_{w_I}]_S. \end{align*} $$

Proof. Let $P\subset G$ be the parabolic subgroup corresponding to I, and let $\mathfrak p=Lie(P)$ . Recall that the tangent bundle $T(G/P)$ has the following description: $T(G/P)=G\times ^P({\mathfrak {g}}/\mathfrak p)$ . Let $\operatorname {\mathrm {pr}}:{\mathfrak {g}}\to {\mathfrak {g}}/\mathfrak p$ denote the projection, and consider the section $s:G/P\to T(G/P)$ given by

$$ \begin{align*} s(gP)=(g,\operatorname{\mathrm{pr}}(Ad(g^{-1})e)). \end{align*} $$

The zero scheme of s is supported at the single point $1.P$ , and S acts on s via the character t. It follows from Corollary 2.4 that there exists an integer N such that

(22) $$ \begin{align} N[1.P]_S = e^S(T(G/P)\otimes\underline{\mathbb{C}}_{-t})\cap [G/P]_S. \end{align} $$

Let $\chi (\_)$ denote the Euler characteristic. Mapping Equation (22) to ordinary cohomology, we obtain

$$ \begin{align*}N = e(T(G/P))\cap [G/P] = \chi(G/P), \end{align*} $$

where the second equality follows from the Poincaré–Hopf index theorem (see, e.g., [Reference Guillemin and Pollack25, Chapter 3]). Moreover, since the Euler characteristic of $G/P$ equals the number of Schubert cells in $G/P$ , we have $N = \frac {|W|}{|W_I|}$ . Pulling back Equation (22) along the (S-equivariant) flat map $\pi :G/B\to G/P$ , see [Reference Edidin and Graham18], we have

$$ \begin{align*} \frac{|W|}{|W_I|}[X_{w_I}]_S=e^S(\pi^*T(G/P)\otimes\underline{\mathbb{C}}_{-t})\cap [G/B]. \end{align*} $$

Finally, we observe that $\pi ^*T(G/P)=G\times ^B{\mathfrak {g}}/\mathfrak p$ has a filtration with quotients , so that

$$ \begin{align*}e^S(\pi^*T(G/P)\otimes \underline{\mathbb C}_{-t}) = \prod_{\alpha\in\Phi^+\backslash\Phi_I^+ } (c^S_1(\mathcal L_\alpha)-t). \end{align*} $$

The result of the proposition now follows after capping with $[G/B]$ .

Using Proposition 6.2, we can compute cohomology classes in $H^*_S(\mathbf {P})$ that are dual to the Peterson subvarieties.

Theorem 6.3. For any $I\subset \Delta $ , we have

$$ \begin{align*} \prod\limits_{\alpha\in\Delta\backslash I}(q_\alpha-2t)\cap[\mathbf{P}]_S=\frac{|W|}{|W_I|}[\mathbf{P}_I]_S. \end{align*} $$

Proof. It is sufficient to prove the result in the case where $I=\Delta \backslash \{\alpha \}$ for some $\alpha \in \Delta $ . Let $G_I$ be the standard Levi subgroup corresponding to $I\subset \Delta $ , and let $B_I=G_I\cap B$ . We identify $G_I/B_I$ with the Schubert variety $X_{w_I}$ . For convenience, we write $r_\alpha =c_1^S(\mathcal L_\alpha )-t$ . We have (in $H^S_*(G/B)$ )

$$ \begin{align*} r_\alpha\cap[\mathbf{P}]_S &=r_\alpha\prod\limits_{\beta\in\Phi^+\backslash\Delta} r_\beta\cap [G/B]_S &&\text{by Equation (13)}\\ &= \prod\limits_{\beta\in\Phi_I^+\backslash I}r_\beta\prod\limits_{\beta\in\Phi^+\backslash\Phi_I^+} r_\beta\cap [G/B]_S\\ &=\frac{|W|}{|W_I|} \prod\limits_{\beta\in\Phi_I^+\backslash I}r_\beta\cap[G_I/B_I]_S &&\text{by {P}roposition~6.2}\\ &=\frac{|W|}{|W_I|}[\mathbf{P}_I]_S &&\text{by Equation (13)}. \end{align*} $$

Following Lemma 6.1, we have $i^*r_\alpha =q_\alpha -2t$ . The result follows from the projection formula (Equation (7)) applied to the inclusion $i:\mathbf {P}\to G/B$ .

As an application of Theorem 6.3, we recover the presentation of $H^*_S(\mathbf {P})$ as a quotient of $H^*_S(G/B)$ , first obtained by Harada, Horiguchi, and Masuda [Reference Harada, Horiguchi and Masuda26].

Corollary 6.4. Recall that $q_\alpha =\sum _{\beta \in \Delta }\left \langle \beta ^\vee ,\alpha \right \rangle p_\beta $ . The equivariant cohomology ring of the Peterson variety admits the presentation

$$ \begin{align*} H^*_S(\mathbf{P})=\frac{\mathbb Q[p_\alpha]_{\alpha\in\Delta}}{\left\langle p_\alpha(q_\alpha-2t)\right\rangle}_{\alpha\in\Delta}. \end{align*} $$

Proof. Following [Reference Drellich17], the map $H^*_S(G/B)\to H^*_S(\mathbf {P})$ is surjective. Recall that $H^*_T(G/B)$ is generated (as a ring) by the divisor classes. Consequently, the are ring generators for $H^*_S(\mathbf {P})$ .

Let $I=\Delta \backslash \{\alpha \}$ . Following Theorem 6.3 and [Reference Goldin, Mihalcea and Singh22, Theorems 6.5 and 6.6], we have

$$ \begin{align*} (q_\alpha-2t)\cap[\mathbf{P}]_S & =\frac{|W|}{|W_I|} [\mathbf{P}_I]_S,\\ p_\alpha\cap [\mathbf{P}_I]_S & =0, \end{align*} $$

and hence $ p_\alpha (q_\alpha -2t)\cap [\mathbf {P}]_S=0. $ By the universal coefficients theorem [Reference May34, Chapter 17, Section 3], the map $\omega \mapsto \omega \cap [\mathbf {P}]_S$ is an isomorphism $H^*_S(\mathbf {P})\xrightarrow \sim H_*^S(\mathbf {P})$ . It follows that $p_\alpha (q_\alpha -2t)=0$ . Consequently, we obtain a surjective map

(23) $$ \begin{align} \frac{\mathbb Q[p_\alpha]_{\alpha\in\Delta}}{\left\langle p_\alpha(q_\alpha-2t)\right\rangle}_{\alpha\in\Delta}\longrightarrow H^*_S(\mathbf{P}). \end{align} $$

It remains to show that this map is an isomorphism, that is, there are no other relations. Proposition 5.3, specialized to the case of the Peterson and permutohedral varieties, states that $H^*(\mathbf {P})\cong H^*(\mathbf {Perm})^W$ . Using Klyachko’s presentation [Reference Klyachko30, Theorem 3] of the latter’s cohomology, we have

$$ \begin{align*} H^*(\mathbf{P})=\dfrac{\mathbb Q[p_\alpha]_{\alpha\in\Delta}}{\left\langle p_\alpha q_\alpha\right\rangle_{\alpha\in\Delta}}. \end{align*} $$

We deduce that the kernel of the map in (23) is t-divisible. Since $H^*_S(\mathbf {P})$ is torsion free over $H^*_S(pt)$ , we conclude that (23) is an isomorphism.

6.2. The Chevalley and Monk formulas

Theorem 6.5 (Equivariant Chevalley formula)

For $\alpha \in \Delta $ , $J\subset \Delta $ , we have

$$ \begin{align*} p_\alpha\cap[\mathbf{P}_J]_S= \begin{cases} 0 &\text{if }\alpha\not\in J,\\ \left\langle2\rho_J^\vee,\varpi^J_\alpha\right\rangle t\,[\mathbf{P}_J]_S+ \sum\limits_{\beta\in J} \left\langle\varpi^{J\vee}_\beta,\varpi^J_\alpha\right\rangle \frac{|W_J|}{|W_{J\backslash \{\beta\}}|}[\mathbf{P}_{J\backslash \{\beta\}}]_S &\text{if }\alpha\in J. \end{cases} \end{align*} $$

Proof. Recall the inclusion map $\iota _J:\mathbf {P}_J\hookrightarrow \mathbf {P}$ . Following [Reference Goldin, Mihalcea and Singh22, Theorem 6.6(b)], we have

$$ \begin{align*}\iota_J^*p_\alpha= \begin{cases} p_\alpha & \text{if }\alpha\in J,\\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

In particular, we have $p_\alpha \cap [\mathbf {P}_J]_S=0$ for $\alpha \not \in J$ .

Next, for $\alpha \in J$ , we have $ \varpi ^J_\alpha = \sum _{\beta \in J} \left \langle \varpi ^{J\vee }_\beta ,\varpi ^J_\alpha \right \rangle \beta , $ and hence

$$ \begin{align*} &&\iota_J^*p_\alpha =\sum_{\beta\in J} \left\langle \varpi^{J\vee}_\beta,\varpi^J_\alpha\right\rangle \iota_J^* q_\beta &&\text{in }H^*_S(\mathbf{P}_J). \end{align*} $$

Following Theorem 6.3, we have

$$ \begin{align*} &&q_\beta\cap[\mathbf{P}_J]_S =\frac{|W_J|}{|W_{J\backslash\{\beta\}}|}[\mathbf{P}_{J\backslash\{\beta\}}]_S+2t[\mathbf{P}_J]_S. && \end{align*} $$

Further, for $\beta \in J$ , we have $\iota _J^*q_\beta =\sum _{\alpha \in J}a_{\beta \alpha }\iota _J^*p_\alpha =\sum _{\alpha \in J}a_{\beta \alpha }p_\alpha =q_\beta $ . Hence, by the projection formula (Equation (7)),

$$ \begin{align*} p_\alpha\cap[\mathbf{P}_J]_S&=\sum_{\beta\in J}\langle \varpi^{J\vee}_\beta,\varpi^J_\alpha\rangle\, q_\beta \cap [\mathbf{P}_J]_S \\ &= \sum_{\beta\in J} \left\langle\varpi^{J\vee}_\beta,\varpi^J_\alpha\right\rangle \left( 2t [\mathbf{P}_J]_S+ \frac{|W_J|}{|W_{J\backslash\{\beta\}}|}[\mathbf{P}_{J\backslash\{\beta\}}]_S\right) \\ &= \left\langle2\rho_J^\vee,\varpi^J_\alpha\right\rangle t [\mathbf{P}_J]_S+ \sum_{\beta\in J} \left\langle\varpi^{J\vee}_\beta,\varpi^J_\alpha\right\rangle \frac{|W_J|}{|W_{J\backslash\{\beta\}}|}[\mathbf{P}_{J\backslash\{\beta\}}]_S. \end{align*} $$

Example 6.6. Let $\Delta =B_2$ .

We use Theorem 6.5 to compute

$$ \begin{align*} p_1\cap[\mathbf{P}]_S= \left\langle2\rho^\vee,\varpi_1\right\rangle t\,[\mathbf{P}]_S+ \left\langle\varpi^{\vee}_1 ,\varpi_1\right\rangle \frac{|W|}{|W_{\{2\}}|}[\mathbf{P}_{\{2\}}]_S+ \left\langle\varpi^{\vee}_2,\varpi_1\right\rangle \frac{|W|}{|W_{\{1\}}|}[\mathbf{P}_{\{1\}}]_S. \end{align*} $$

Recall the realization of the root system of $B_2$ in $\mathbb R^2$ , given by $\alpha _1=\epsilon _1-\epsilon _2$ , $\alpha _2=\epsilon _2$ . We have $\varpi _1=\epsilon _1=\alpha _1+\alpha _2$ . Observe that $\langle \varpi _j^\vee ,\varpi _i\rangle $ equals the coefficient of $\alpha _j$ in the expansion of $\varpi _i$ as a sum of simple roots. Hence, we have

$$ \begin{align*} \left\langle\varpi_1^\vee,\varpi_1\right\rangle=1,&& \left\langle\varpi_2^\vee,\varpi_1\right\rangle=1. \end{align*} $$

Furthermore, $2\rho ^\vee =2\varpi _1^\vee +2\varpi _2^\vee $ , and hence $\left \langle 2\rho ^\vee ,\varpi _1\right \rangle =4$ . Finally, we have $|W|=8$ and $|W_{\{1\}}|=|W_{\{2\}}|=2$ . Therefore,

$$ \begin{align*} p_1\cap[\mathbf{P}]_S=4t[\mathbf{P}]_S+4[\mathbf{P}_{\{1\}}]_S+4[\mathbf{P}_{\{2\}}]_S. \end{align*} $$

Recall that $\Omega _I=\prod _{\alpha \in I}p_\alpha $ , so that, in particular, $\Omega _\alpha =p_\alpha $ . Following Proposition 4.3 and Theorem 5.5, the set is a basis of $H^*_S(\mathbf {P})$ over $\mathbb Q[t]$ . Using Proposition 4.3 and Theorem 5.7, we can deduce a Monk rule for this basis from the Chevalley formula.

Theorem 6.7 (Equivariant Monk rule)

Let $f_J$ denote the connection index of the Dynkin diagram J, that is, the determinant of the Cartan matrix of J. For $\alpha \in \Delta $ , we have

$$ \begin{align*} \Omega_\alpha \Omega_I= \begin{cases} \Omega_{I\cup\{\alpha\}} &\text{if }\alpha\not\in I,\\ 2\left\langle\rho^\vee_I,\varpi^I_\alpha\right\rangle t \Omega_I+ \sum\limits_{\gamma\in\Delta\backslash I} \dfrac{f_{I\cup\{\gamma\}}}{f_I}\left\langle\varpi_\gamma^{(I\cup\{\gamma\})\vee},\varpi^{I\cup\{\gamma\}}_\alpha\right\rangle \Omega_{I\cup\{\gamma\}} &\text{if }\alpha\in I. \end{cases} \end{align*} $$

Proof. Consider the coefficients $c_{\alpha I}^J\in \mathbb Q[t]$ in the product $\Omega _\alpha \Omega _I=\sum c_{\alpha I}^J \Omega _J$ . Following Proposition 4.3, we have

$$ \begin{align*} c_{\alpha I}^J &= \frac{\left\langle \Omega_\alpha \Omega_I,[\mathbf{P}_J]_S\right\rangle} {\left\langle \Omega_J,[\mathbf{P}_J]_S\right\rangle} =\frac{\left\langle \Omega_I,\Omega_\alpha\cap [\mathbf{P}_J]_S\right\rangle} {\left\langle \Omega_J,[\mathbf{P}_J]_S\right\rangle} =\frac{f_J}{|W_J|}\left\langle \Omega_I,\Omega_\alpha\cap [\mathbf{P}_J]_S\right\rangle, \end{align*} $$

where the final equality is from Lemma 5.6.

Consider first $\alpha \not \in J$ . Following Theorem 6.5, we have $\Omega _\alpha \cap [\mathbf {P}_J]_S=0$ , and hence $c_{\alpha I}^J=0$ .

Consider now $\alpha \in J$ . Recall from Proposition 4.3 that $\left \langle \Omega _I,[\mathbf {P}_K]_S\right \rangle =0$ unless $I=K$ . Further, by Theorem 6.5, the only $[\mathbf {P}_I]_S$ appearing in the expansion of $\Omega _\alpha \cap [\mathbf {P}_J]_S$ correspond to $I=J$ or $I=J\backslash \{\gamma \}$ for some $\gamma \in J$ . Thus, $c_{\alpha I}^J=0$ unless $I=J$ or $I=J\backslash \{\gamma \}$ for some $\gamma \in J$ . For $J=I$ , we have

$$ \begin{align*} c_{\alpha I}^I &=\frac{ f_I}{|W_I|}\left\langle\Omega_I, \Omega_\alpha\cap [\mathbf{P}_I]_S\right\rangle\\ &=\frac{f_I}{|W_I|}\left\langle \Omega_I, \left\langle2\rho_I^\vee,\varpi^I_\alpha\right\rangle t\,[\mathbf{P}_I]_S + \sum\limits_{\beta\in I} \left\langle\varpi^{I\vee}_\beta,\varpi^I_\alpha\right\rangle\frac{|W_I|}{|W_{I\backslash\{\beta\}}|}[\mathbf{P}_{I\backslash\{\beta\}}]_S \right\rangle\\ &=\frac{f_I}{|W_I|}\left\langle2\rho_I^\vee,\varpi^I_\alpha\right\rangle t\,\left\langle \Omega_I,[\mathbf{P}_I]_S\right\rangle\\ &= \left\langle2\rho_I^\vee,\varpi^I_\alpha\right\rangle t. \end{align*} $$

In the case where $J=I\sqcup \{\gamma \}$ for some $\gamma \in \Delta $ , we have

$$ \begin{align*} c_{\alpha I}^J&= \frac{ f_J}{|W_J|}\left\langle \Omega_I,\Omega_\alpha\cap [\mathbf{P}_J]\right\rangle\\ &=\frac{f_J}{|W_J|}\left\langle \Omega_I, \left\langle2\rho_J^\vee,\varpi^J_\alpha\right\rangle t\,[\mathbf{P}_J]_S + \sum\limits_{\beta\in J} \left\langle\varpi^{J\vee}_\beta,\varpi^J_\alpha\right\rangle\frac{|W_J|}{|W_{J\backslash\{\beta\}}|}[\mathbf{P}_{J\backslash\{\beta\}}]_S\right\rangle \\ &=\frac{f_J}{|W_J|}\frac{|W_J|}{|W_I|} \left\langle\varpi^{J\vee}_\gamma,\varpi^J_\alpha\right\rangle \left\langle \Omega_I, [\mathbf{P}_I]_S \right\rangle \\ &= \left\langle\varpi^{J\vee}_\gamma,\varpi^J_\alpha\right\rangle \frac{f_J}{f_I}. \end{align*} $$

Example 6.8. Consider $\Delta =B_3$ , and let $I=\{1,2\}\subset \Delta $ .

We compute the product $\Omega _{2}\Omega _I$ . By Theorem 6.7,

$$ \begin{align*} \Omega_{2}\Omega_I&= 2\left\langle\rho^\vee_I,\varpi_{2}^I\right\rangle t \Omega_I+ \sum\limits_{\gamma\in\Delta\backslash I} \dfrac{f_{I\cup\{\gamma\}}}{f_I}\left\langle\varpi_\gamma^{(I\cup\{\gamma\})\vee},\varpi^{I\cup\{\gamma\}}_{2}\right\rangle \Omega_{I\cup\{\gamma\}}\\ &= 2\left\langle\rho^\vee_I,\varpi_{2}^I\right\rangle t \Omega_I+ \dfrac{f_{\Delta}}{f_I} \left\langle\varpi_{3}^{\vee},\varpi^{}_{2}\right\rangle \Omega_\Delta. \end{align*} $$

Since the subdiagram I is isomorphic to $A_2$ , the term $\left \langle \rho _I^\vee ,\varpi _2^I\right \rangle $ is calculated in $A_2$ . We have $\rho _I^\vee =\frac 12(\alpha _1^\vee +\alpha _2^\vee +(\alpha _1^\vee +\alpha _2^\vee ))=\alpha _1^\vee +\alpha _2^\vee $ , and hence $\langle \rho _I^\vee ,\varpi _2^I\rangle =1$ .

The term $\left \langle \varpi _3^\vee ,\varpi _2\right \rangle $ is the coefficient of $\alpha _3$ in the expansion of $\varpi _2$ as a sum of simple roots. Recall the usual realization of the $B_3$ root system inside a three-dimensional vector space with orthonormal basis $\left \{\epsilon _1,\epsilon _2,\epsilon _3\right \}$ , given by $\alpha _1=\epsilon _1-\epsilon _2$ , $\alpha _2=\epsilon _2-\epsilon _3$ , and $\alpha _3=\epsilon _3$ . The fundamental weight $\varpi _2$ is given by

$$ \begin{align*} \varpi_2=\epsilon_1+\epsilon_2=\alpha_1+2\alpha_2+2\alpha_3, \end{align*} $$

and hence $\left \langle \varpi _3^\vee ,\varpi _2\right \rangle =2$ . The connection indices are

$$ \begin{align*} f_I=\det\begin{pmatrix}2&-1\\-1&2\end{pmatrix}=3,&& f_\Delta=\det\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\\\end{pmatrix}=2. \end{align*} $$

Hence, we have $\Omega _2\Omega _I=2t\Omega _I+\frac 43\Omega _\Delta $ .

Remark 6.9. Fix a Coxeter element $v_I$ for each $I\subset \Delta $ , and set $p_{v_I}=i^*\sigma _{v_I}^S$ . It is common in the literature to work with the basis . We see from Theorem 5.5 that and are related by a diagonal change of basis matrix. This allows us to translate Theorem 6.7 into a Monk rule for the basis ,

$$ \begin{align*} p_\alpha p_{v_I}= \begin{cases} \dfrac{(|I|+1)R(v_I)}{R(v_{I\cup\{\alpha\}})}p_{v_{I\cup\{\alpha\}}} &\text{if }\alpha\not\in I,\\ 2 \left\langle\rho^\vee_I,\varpi_\alpha\right\rangle t p_{v_I}+ \sum\limits_{\gamma\in\Delta\backslash I} \dfrac{(|I|+1)f_{I\cup\{\gamma\}} R(v_I)}{f_I R(v_{I\cup\{\gamma\}})} \left\langle\varpi_\gamma^{I\cup\{\gamma\}\vee},\varpi^{I\cup\{\gamma\}}_\alpha\right\rangle p_{v_{I\cup\{\gamma\}}} &\text{if }\alpha\in I. \end{cases} \end{align*} $$

6.3. Tables of structure constants

Observe that $\left \langle \varpi _\gamma ^{J\vee },\varpi _\alpha ^J\right \rangle $ is precisely the coefficient of $\gamma $ in the expression of the fundamental weight $\varpi _\alpha ^J$ as a sum of the simple roots in J. These coefficients, and the connection indices of the Dynkin diagrams, are listed in [Reference Onishchik and Vinberg36, Tables 2 and 3] (see also [Reference Bourbaki11, Chapter 6, Section 4]). Using this, we can compute the structure constants in the Chevalley and Monk formulas. For the reader’s convenience, we tabulate the equivariant structure constants for the Monk and Chevalley formulas in types A–D. The ordinary structure constants in the Monk rule for all Dynkin diagrams have also been recently computed and tabulated by Horiguchi in [Reference Horiguchi28, Table 2].Footnote ¹

We will denote by $c_{iJ}^K$ and $d_{iJ}^K$ the structure constants given by

$$ \begin{align*} \Omega_{\alpha_i}\Omega_I=\sum c_{iI}^J\Omega_J,&& p_{\alpha_i}\cap[\mathbf{P}_J]_S=\sum d_{iJ}^K[\mathbf{P}_K]_S, \end{align*} $$

respectively. We write i for $\alpha _i\in I$ to simplify notation. Following Theorems 6.5 and 6.7, for $i\in I$ , we have

$$ \begin{align*} &&c_{iI}^I & =d_{iI}^I =\left\langle2\rho_I^\vee,\varpi^I_i\right\rangle t, & \\ &&c_{iI}^J & = \dfrac{f_J}{f_I}\left\langle\varpi_j^{J\vee},\varpi^J_i\right\rangle & J=I\sqcup\{j\}, \end{align*} $$

and for $i\in J$ , we have

$$ \begin{align*} &&d_{iJ}^K=\left\langle\varpi^{J\vee}_j,\varpi^J_i\right\rangle \frac{|W_J|}{|W_K|},&& K=J\backslash\{j\}. \end{align*} $$

For the classical type Dynkin diagrams, these values are listed in Table 1. The ordinary structure constants for the Chevalley formula are listed in Table 2. The structure constants not involving t, in the Monk rule, are listed (for types A–D) in Table 3.

Table 1 Structure constant for the leading term in the Monk and Chevalley formulas

Table 2 Structure constants for the Chevalley formula

Table 3 Ordinary terms in the Monk rule (see Theorem 6.7)

Example 6.10. Consider $\Delta =B_3$ and $I=\{2,3\}$ . We compute the product $p_2\cap [\mathbf {P}_I]_S$ using Tables 1 and 2.

We have $ p_2\cap [\mathbf {P}_I]_S=d_{2I}^I[\mathbf {P}_I] +d_{2I}^{\{2\}}[\mathbf {P}_{\{2\}}]_S+d_{2I}^{\{3\}}[\mathbf {P}_{\{3\}}]_S. $ The subdiagram I is isomorphic to $B_2$ , so the coefficient $d_{2I}^I=4t$ corresponds to $i=1$ , $n=2$ in the second row of Table 1. The coefficient $d_{2I}^{\{2\}}=4$ corresponds to $i=1$ , $j=2$ , and $n=2$ in the second row of Table 2. The coefficient $d_{2I}^{\{3\}}=4$ corresponds to $i=1$ , $j=1$ , and $n=2$ in the second row of Table 2. Hence, we have

$$ \begin{align*} p_2\cap[\mathbf{P}_I]_S=4t[\mathbf{P}_I]_S+4[\mathbf{P}_{\{2\}}]_S+4[\mathbf{P}_{\{3\}}]_S. \end{align*} $$

Recall from Theorem 6.5 that the coefficients in the Chevalley formula do not depend on the superset $\Delta $ containing I. This phenomenon can be observed by comparing Example 6.6 with Example 6.10.

Example 6.11. Consider $\Delta =D_6$ , and let $I=\{3,4,5\}\subset \Delta $ .

We compute the product $\Omega _{3}\Omega _I$ using Tables 1 and 3. Observe first that for $\gamma \in \Delta \backslash I$ , $J=I\cup \{\gamma \}$ , we have $\left \langle \varpi _\gamma ^{J\vee },\varpi _\alpha ^J\right \rangle =0$ if $\gamma $ is not connected to I. We deduce that $c_{\alpha _3I}^{I\cup \{1\}}=0$ , and that

$$ \begin{align*} \Omega_{3}\Omega_I &=c_{3I}^I\Omega_I+c_{3I}^{I\cup\{2\}}\Omega_{I\cup\{2\}}+c_{3I}^{I\cup\{6\}}\Omega_{I\cup\{6\}}. \end{align*} $$

The coefficient $c_{3I}^I=3t$ corresponds to $i=1$ and $n=3$ in the first row of Table 1, the coefficient $c_{3I}^{I\cup \{6\}}=\frac 12$ corresponds to $i=1$ and $n=4$ in the sixth row of Table 3, and the coefficient $ c_{3I}^{I\cup \{2\}}=\frac 34$ corresponds to $i=3$ and $n=4$ in the first row of Table 3. Hence,

$$ \begin{align*} \Omega_{3}\Omega_I=3t\Omega_I+\frac34\Omega_{I\cup\{2\}}+\frac12\Omega_{I\cup\{6\}}. \end{align*} $$