2.1 Permutations

With some modifications, we follow [Reference Lam, Lee and Shimozono21] for permutations.

We write $\operatorname {\mathrm {Bij}}( X )$ for the group of all bijections of a set X to itself. We will only consider subsets $X\subseteq \mathbb {Z}$ , and we focus on the subgroup $\mathcal {S}_{\mathbb {Z}} \subseteq \operatorname {\mathrm {Bij}}(\mathbb {Z})$ consisting of all w such that $\{ i\in \mathbb {Z}\,|\, w(i)\neq i \}$ is finite – that is, w fixes all but finitely many integers. Some variations will be discussed in §6.

The subgroup $\mathcal {S}_{\neq 0}$ is $\mathcal {S}_+\times \mathcal {S}_-$ , where $\mathcal {S}_+ = \mathcal {S}_{\mathbb {Z}} \cap \operatorname {\mathrm {Bij}}(\mathbb {Z}_{>0})$ and $\mathcal {S}_- = \mathcal {S}_{\mathbb {Z}} \cap \operatorname {\mathrm {Bij}}(\mathbb {Z}_{\leq 0})$ . That is, $\mathcal {S}_{\neq 0}$ is the subgroup of $\mathcal {S}_{\mathbb {Z}}$ preserving the subsets of positive and non-positive integers.

For finite intervals $[m,n]$ , we usually write $\mathcal {S}_{[m,n]} = \operatorname {\mathrm {Bij}}([m,n])$ , and $\mathcal {S}_n = \mathcal {S}_{[1,n]}$ for $n>0$ . We have

$$\begin{align*}\mathcal{S}_+ = \bigcup_{n>0} \mathcal{S}_{[1,n]}, \quad \mathcal{S}_- = \bigcup_{n>0} \mathcal{S}_{[-n,0]}, \quad \text{and} \quad \mathcal{S}_{\mathbb{Z}} = \bigcup_{n>0} \mathcal{S}_{[-n,n]}. \end{align*}$$

Elements $w\in \mathcal {S}_{\mathbb {Z}}$ are written in one-line notation: choose an interval $[m,n]$ so that $w(i)=i$ for all i outside $[m,n]$ , and write $w=[w(m),\ldots ,w(n)]$ .

Bruhat order on $\mathcal {S}_{\mathbb {Z}}$ is defined as follows. For each $p,q\in \mathbb {Z}$ and $w\in \mathcal {S}_{\mathbb {Z}}$ , we set

$$\begin{align*}k_w(p,q) = \# \{ a\leq p \,|\, w(a)> q \}. \end{align*}$$

Then $v\leq w$ in $\mathcal {S}_{\mathbb {Z}}$ if $k_v(p,q)\leq k_w(p,q)$ for all $p,q\in \mathbb {Z}$ .

An element $w\in \mathcal {S}_{\mathbb {Z}}$ is Grassmannian if it has no descents except possibly at $0$ , so $w(i)<w(i+1)$ for all $i\neq 0$ . Grassmannian elements are in correspondence with partitions $\lambda $ : given a Grassmannian permutation w, the partition $\lambda = (\lambda _1\geq \lambda _2 \geq \cdots \geq 0)$ is defined by $\lambda _k = w(1-k) -1+k$ , for $k>0$ . Conversely, given $\lambda $ , one defines $w=w_\lambda $ by setting $w_\lambda (k)=\lambda _{1-k}+k$ for $k\leq 0$ , and then filling in the positive values with the unused integers in increasing order.

The length $\ell (w)$ of $w\in \mathcal {S}_{\mathbb {Z}}$ is the cardinality of the (finite) set $\{i<j\,|\, w(i)>w(j)\}$ .

The element $w_\circ ^\infty \in \operatorname {\mathrm {Bij}}(\mathbb {Z})$ defined by $w_\circ ^\infty (i) = 1-i$ does not lie in $\mathcal {S}_{\mathbb {Z}}$ , but conjugation by $w_\circ ^\infty $ defines a length-preserving outer automorphism $\omega $ of $\mathcal {S}_{\mathbb {Z}}$ :

$$\begin{align*}\omega(w)(i) = (w_\circ^\infty w w_\circ^\infty)(i) = 1-w(1-i). \end{align*}$$

2.2 Vector spaces

Let V be a countable-dimensional vector space with basis $e_i$ for $i\in \mathbb {Z}$ . For any interval $[m,n]$ , there is a subspace $V_{[m,n]}$ with basis $e_i$ for $i\in [m,n]$ . For semi-infinite intervals we usually write $V_{\leq n}$ , or $V_{>m}$ . The standard flag $V_{\leq \bullet }$ in V has components $V_{\leq k}$ with basis $e_i$ for $i\leq k$ , for each $k\in \mathbb {Z}$ . The opposite flag $V_{>\bullet }$ is comprised of spaces ${V}_{>k}$ spanned by $e_i$ for $i>k$ . Clearly, $V=V_{\leq 0} \oplus V_{>0}$ (and $V=V_{\leq k} \oplus V_{>k}$ for any k).

When the context is clear, we use the same notation for standard and opposite flags in $V_{(m,n]}$ – for instance, writing $V_{\leq k} \subseteq V_{(m,n]}$ instead of $V_{(m,k]} \subseteq V_{(m,n]}$ .

A torus T acts on V, so that $e_i$ is scaled by the character $y_i$ , for $i\in \mathbb {Z}$ . So T also acts on each subspace $V_{[m,n]}$ . We generally take T to be the countable product $T=\prod _{i\in \mathbb {Z}} \mathbb {C}^*$ , so that its classifying space is $\prod _{i\in \mathbb {Z}} \mathbb {P}^\infty $ . This is an inverse limit of finite products of $\mathbb {P}^\infty $ , so the T-equivariant cohomology of a point is a polynomial ring in the y variables:

$$\begin{align*}H_T^*(\mathrm{pt}) = \mathbb{Z}[y] = \mathbb{Z}[\ldots,y_{-1},y_0,y_1,\ldots]. \end{align*}$$

(For those who prefer finite dimensional groups, one may also take T to be any torus, with weights $y_i$ , for $i\in \mathbb {Z}$ . By taking T sufficiently large, any given finite set of y’s can be made algebraically independent.)

2.3 Flag varieties

For any vector space W, the flag variety $ {Fl} _+(W)$ is the space of all complete flags of finite-dimensional subspaces of W. That is, a point of $ {Fl} _+(W)$ is $E_\bullet = (0 \subset E_1 \subset E_2 \subset \cdots \subset W)$ , where $\dim E_i = i$ . When W is finite-dimensional, this is the usual complete flag variety. In general, it is a limit of finite-dimensional flag varieties: to construct $ {Fl} _+(W)$ , for each $d>0$ , one forms $ {Gr} (d,W)$ as the union of $ {Gr} (d,U)$ over finite-dimensional subspaces $U\subset W$ ; then $ {Fl} _+(W)$ embeds naturally in the product $\prod _{d>0} {Gr} (d,W)$ . So $ {Fl} _+(W)$ inherits its topology from the product topology on the Grassmannians. This is the same as the inverse limit topology with respect to projections onto partial flag varieties.

There is also a variety $ {Fl} _-(W)$ parametrizing flags of finite-codimensional subspaces of W, but here an extra requirement is imposed: one fixes a flag $W^\bullet $ of finite-codimensional subspaces of W. Then a point of $ {Fl} _-(W)$ is $E^\bullet = ( \cdots \subset E^2 \subset E^1 \subset W)$ , where $E^i$ has codimension i in W, and each $E^i$ contains some $W^j$ . (Often we negate indices and write $E_{-i}=E^i$ for such flags.) Equivalently, let $K_i = W/W^i$ , and consider the restricted dual space $W^{*'} = \bigcup _i K_i^*$ . (This is finite-dimensional when W is, and countable-dimensional if $\dim W$ is infinite.) Then $ {Fl} _-(W) = {Fl} _+(W^{*'})$ .

In our setting, an equivalent construction of these varieties is as follows. The flag variety $ {Fl} (1,\ldots ,n;V_{>0})$ is a union of finite-dimensional partial flag varieties $ {Fl} (1,\ldots ,n;V_{[1,m]})$ over $m\geq n$ , with respect to standard embeddings coming from $V_{[1,m]} \subset V_{[1,m+1]}$ .

The finite-dimensional flag varieties have tautological bundles $S_i$ , and T acts, restricting its action on V. Taking the graded inverse limit of cohomology rings, one has

$$\begin{align*}H_T^* {Fl} (1,\ldots,n;V_{>0}) = \mathbb{Z}[y][x_1,\ldots,x_n], \end{align*}$$

where $x_i$ restricts to $-c_1^T(S_i/S_{i-1})$ on each finite-dimensional variety.

Next we take the inverse limit of $ {Fl} (1,\ldots ,n;V_{>0})$ over n, using natural projections. (So it is a ‘pro-ind-variety’: the inverse limit of a direct limit of algebraic varieties.) Its equivariant cohomology is the direct limit of rings $\mathbb {Z}[y][x_1,\ldots ,x_n]$ as $n\to \infty $ , so

$$\begin{align*}H_T^* {Fl} _+(V_{>0}) = \mathbb{Z}[y][x_1,x_2,\ldots]. \end{align*}$$

Similarly, the construction of $ {Fl} _-(V_{\leq 0})$ (with respect to the standard flag $V_{\leq \bullet }$ ) realizes it as a limit of the flag varieties $ {Fl} (m-n,\ldots ,m;V_{(-m,0]})$ , which have tautological bundles $S_i$ of codimension $-i$ , for $i\leq 0$ . Its equivariant cohomology is

$$\begin{align*}H_T^* {Fl} _-(V_{\leq0}) = \mathbb{Z}[y][x_0,x_{-1},\ldots], \end{align*}$$

where again $x_i$ restricts to $-c_1^T(S_{i}/S_{i+1})$ on each finite-dimensional variety, for $i\leq 0$ .

Remark 2.1. One sometimes sees yet another limit, taking a union $\bigcup _{n>0} {Fl} (V_{[1,n]})$ over the standard embeddings $V_{[1,n]} \subset V_{[1,n+1]}$ . This leads to what might be called a restricted flag variety $ {Fl} ^{\prime }_+(V_{>0})$ , parametrizing flags $E_\bullet $ of finite-dimensional subspaces which are eventually standard: $E_k = V_{\leq k}$ for all $k \gg 0$ . As a direct limit, its cohomology is

$$\begin{align*}H_T^* {Fl} ^{\prime}_+(V_{>0}) = \mathbb{Z}[y][\![x]\!]_{\mathrm{gr}}, \end{align*}$$

the ring of graded power series in x with coefficients in y. (For example, the infinite sum $\sum _{i>0} x_i$ is an element of this ring.) The embedding $ {Fl} ^{\prime }_+(V_{>0}) \hookrightarrow {Fl} _+(V_{>0})$ corresponds to the inclusion of the polynomial ring $\mathbb {Z}[y][x] \hookrightarrow \mathbb {Z}[y][\![x]\!]_{\mathrm {gr}}$ .

We will not make use of these restricted varieties, except to mention their appearance in the literature. One of several advantages of working with $ {Fl} _+(V_{>0})$ rather than $ {Fl} ^{\prime }_+(V_{>0})$ is that elements of its cohomology are automatically polynomials.

2.4 A technical note on limits

For a rising union of spaces $X= \bigcup X_n$ , the direct limit topology is defined so that a subset $U\subset X$ is open exactly when each intersection $U\cap X_n$ is open. For an inverse system of spaces $\cdots \to X_n \to X_{n-1} \to \cdots $ , the inverse limit topology on $X= \varprojlim X_n$ is the coarsest topology so that the projections $X \to X_n$ are continuous; in our context, this is a subspace of the product topology on $\prod X_n$ .

From the contravariance of cohomology, one may naively expect that

$$\begin{align*}H^*\left( \bigcup X_n \right) = \varprojlim H^*(X_n) \quad \text{ and } \quad H^*\left( \varprojlim X_n \right) = \varinjlim H^*(X_n). \end{align*}$$

Using Čech-Alexander-Spanier cohomology, and for the relatively nice topological spaces we encounter, these naive expectations hold. For finite-dimensional algebraic varieties, this cohomology theory agrees with the more familiar singular cohomology. These facts may be gleaned from standard algebraic topology texts; see also [Reference Anderson and Fulton6, Appendix A].

7.1 Duality

Fix a linear isomorphism $f\colon V \xrightarrow {\sim } V^{*'}$ , where as before, $V^{*'} \subset V^*$ is the restricted dual defined with respect to our chosen flag $V_{\leq {\scriptscriptstyle {\bullet }} }$ . For any subspace $E\subseteq V$ , one has the associated orthogonal complement

$$\begin{align*}E^\perp = \{ v\in V \,|\, f(u)(v) = 0 \text{ for all } u \in E \}. \end{align*}$$

This operation reverses inclusion, so the image of the standard flag is given by the spaces $V^\perp _{\leq -k}$ .

There is a duality morphism

$$\begin{align*}\mathrm{Gr}^k(V; V_{\leq\bullet} ) \to \mathrm{Gr}^{-k}(V; V^\perp_{\leq-\bullet}), \end{align*}$$

by $E \mapsto E^\perp $ .

The same formula defines an automorphism of $\mathrm {Fl}(V)$ , sending a flag with components $E_k$ to one with components $E_{-k}^\perp $ .

From now on, we assume the isomorphism $f\colon V \to V^{*'}$ is given by the skew-symmetric form sending $e_i \mapsto e_{1-i}^*$ for $i>0$ , and $e_i \mapsto -e_{1-i}^*$ for $i\leq 0$ . In this case, the duality morphism is an involution, equivariant with respect to the automorphism of T defined on characters by $y_i \mapsto -y_{1-i}$ , and the standard flag is preserved, with $(V_{\leq k})^\perp = V_{\leq -k}$ . (All of this holds as well for a symmetric form.)

The induced automorphism $\omega $ of $H_T^*\mathrm {Fl} =\Lambda [x;y]$ is given by

$$\begin{align*}\omega(c) = 1/(1-c_1+c_2-\cdots), \quad \omega(x_i) = -x_{1-i}, \quad \omega(y_i)=-y_{1-i}. \end{align*}$$

The same notation is used for the automorphism of $\mathcal {S}_{\mathbb {Z}}$ , defined by $\omega (w)(i) = 1-w(1-i)$ . One checks that $k_{\omega (w)}(p,q) = k_w(-p,-q)$ , so the duality morphism sends $\Omega _w$ to $\Omega _{\omega (w)}$ . It follows that

$$\begin{align*}\omega(\mathbf{S}_w(c;x;y)) = \mathbf{S}_{\omega(w)}(c;x;y). \end{align*}$$

Following [Reference Lam, Lee and Shimozono21], one defines $\mathfrak {S}_w(x;y)$ for any $w\in \mathcal {S}_{\neq 0}$ using the duality involution: for $w=w_-\cdot w_+$ , with $w_-\in \mathcal {S}_-$ and $w_+\in \mathcal {S}_+$ , one defines $\mathfrak {S}_w = \omega (\mathfrak {S}_{\omega (w_-)})\cdot \mathfrak {S}_{w_+}$ .

7.2 Projections

For each k, there is a projection $\pi _k\colon \mathrm {Fl} \to \mathrm {Gr}^k$ , sending $E_\bullet $ to $E_k$ . This is a fiber bundle, and the fiber over $V_{\leq k}\in \mathrm {Gr}^k$ is $ {Fl} _-(V_{\leq k})\times {Fl} _+(V_{>k})$ . In particular, the inclusion $\Lambda [y] \hookrightarrow \Lambda [x;y]$ corresponds to $\pi _0^*$ , and the homomorphism

$$\begin{align*}\Lambda[x;y] \to \mathbb{Z}[x;y], \qquad c\mapsto 1 \end{align*}$$

corresponds to restriction to the fiber over $V_{\leq 0}\in \mathrm {Gr}$ .

7.3 Shift

Let $\operatorname {\mathrm {sh}} \colon V \to V$ be the linear automorphism given by $e_i \mapsto e_{i-1}$ . This induces shift morphisms, also written $\operatorname {\mathrm {sh}}\colon \mathrm {Gr}^{k} \to \mathrm {Gr}^{k-1}$ , sending $E\subset V$ to $\operatorname {\mathrm {sh}}(E)\subset V$ , and an automorphism $\operatorname {\mathrm {sh}}\colon \mathrm {Fl} \to \mathrm {Fl}$ , defined by $\operatorname {\mathrm {sh}}(E_\bullet )_k = \operatorname {\mathrm {sh}}(E_{k+1})$ . The shift morphisms are equivariant with respect to a similar automorphism of $T = \prod _{i\in \mathbb {Z}} \mathbb {C}^*$ , sending $z_i \mapsto z_{i-1}$ .

To construct the shift morphism from finite-dimensional varieties, one uses the system of maps

$$ \begin{align*} {Gr} (m+k, V_{(-m,m]}) &\hookrightarrow {Gr} (m+k,V_{(-m-1,m+1]}) \\ (E\subset V_{(-m,m]}) &\mapsto ( \operatorname{\mathrm{sh}}(E) \subset V_{(-m-1,m+1]}). \end{align*} $$

Taking the union over m on each side determines a morphism $\mathrm {Gr}^k \rightarrow \mathrm {Gr}^{k-1}$ .

Pullback by the shift morphism gives the translation operator $\gamma \colon \Lambda [x;y] \to \Lambda [x;y]$ on cohomology. Explicitly, $\gamma =\operatorname {\mathrm {sh}}^*$ is given by

$$ \begin{align*} \gamma(x_i) &= x_{i+1},\\ \gamma(y_i) &= y_{i+1}, \text{ and}\\ \gamma(c_k) &= \sum_{p=0}^k c_p\,x_1^{k-p} + y_1\sum_{p=0}^{k-1} c_{p}\,x_1^{k-1-p}. \end{align*} $$

(The action on c variables can be written concisely as $\gamma (c) = c\cdot \frac {1+y_1}{1-x_1}$ .) The action on x variables comes from $\operatorname {\mathrm {sh}}^*(S_i) = S_{i+1}$ , and the y variables are determined by the automorphism of T. For the c variables, one observes $\operatorname {\mathrm {sh}}^*(V_{\leq 0}) = V_{\leq 1}$ , so

$$\begin{align*}\operatorname{\mathrm{sh}}^*c^T(V_{\leq0}-S_0) = c^T(V_{\leq 1} - S_1) = c^T(V_{\leq0}-S_0)\cdot c^T(\mathbb{C}\cdot e_1 - S_1/S_0). \end{align*}$$

The homomorphism $\gamma $ is invertible. For any $m\in \mathbb {Z}$ , one has $\gamma ^m(x_i)=x_{i+m}$ and $\gamma ^m(y_i)=y_{i+m}$ , with the action on c variables determined by

$$ \begin{align*} \gamma^m(c) &= \begin{cases} c\cdot \prod_{i=1}^m\frac{ 1+y_i }{ 1-x_i } & \text{if } m\geq 0; \\ c\cdot \prod_{i=m+1}^0\frac{ 1-x_i }{ 1+y_i } & \text{if } m< 0. \end{cases} \end{align*} $$

For any $w\in \operatorname {\mathrm {Inj}}(\mathbb {Z})$ , the injection $\gamma ^m(w)$ is defined by $\gamma ^m(w)(i) = m+w(i-m)$ .

8.1 Grassmannians

We will describe the setup and state some results for the Grassmannian first, and prove the more general analogues for the flag variety in the following subsection.

As before, there is an isomorphism $H_T^*\mathrm {Gr}(\mathbb {V}) = \Lambda [y]$ . Here we use the notation $\Lambda = \mathbb {Z}[\mathbf {c}]=\mathbb {Z}[\mathbf {c}_1,\mathbf {c}_2,\ldots ]$ , and the map identifies $\mathbf {c}_k = c^T(\mathbb {V}_{\leq 0} - \mathbb {S}_0)$ , where $\mathbb {S}_0$ is the tautological bundle on $\mathrm {Gr}(\mathbb {V})$ . Similarly, one has $H_T^*\mathrm {Fl}(\mathbb {V}) = \Lambda [x;y]$ , with $x_i = -c_1^T( \mathbb {S}_i/\mathbb {S}_{i-1} )$ .

The direct sum morphism

$$\begin{align*}\boxplus\colon \mathrm{Gr}^k(V;V_{\leq\bullet}) \times \mathrm{Gr}^l(V;V_{\leq\bullet}) \to \mathrm{Gr}^{k+l}(\mathbb{V};\mathbb{V}_{\leq\bullet}) \end{align*}$$

given by $\boxplus ( E,F ) = E\oplus F$ is readily checked to be well-defined and T-equivariant.

Proposition 8.1. The morphism

$$\begin{align*}f\colon \mathrm{Gr}(V) \to \mathrm{Gr}(\mathbb{V}), \qquad E \mapsto V_{\leq 0} \oplus E \end{align*}$$

induces the standard isomorphism $\Lambda [y] \to \Lambda [y]$ on cohomology rings, sending $\mathbf {c}_k \mapsto c_k$ .

Proposition 8.2. The homomorphism

$$\begin{align*}H_T^*\mathrm{Gr}(\mathbb{V}) \xrightarrow{\boxplus^*} H_T^*(\mathrm{Gr}(V)\times \mathrm{Gr}(V)) \end{align*}$$

is identified with the homomorphism of $\mathbb {Z}[y]$ -algebras

$$\begin{align*}\Lambda[y] = \mathbb{Z}[\mathbf{c},y] \xrightarrow{\Delta} \Lambda[y]\otimes_{\mathbb{Z}[y]} \Lambda[y] = \mathbb{Z}[c,c',y], \end{align*}$$

given by $\mathbf {c}_k \mapsto c_k + c_{k-1} c^{\prime }_1 + \cdots c_1 c^{\prime }_{k-1} + c^{\prime }_k$ . (Here $c=c^T(V_{\leq 0}-S_0)$ comes from the first factor of $\mathrm {Gr}(V)$ , and $c'=c^T(V_{\leq 0}-S_0')$ comes from the second factor, so $\mathbf {c}=c\cdot c'$ .)

The first of these propositions follows from the second, after replacing c by $c'$ , since $f(E) = \boxplus (E,V_{\leq 0})$ . And the second proposition is simply the equation $\boxplus ^*c^T(\mathbb {V}_{\leq 0} - \mathbb {S}_0) = c^T(V_{\leq 0}+V_{\leq 0}-S_0-S^{\prime }_0) = c^T(V_{\leq 0}-S_0)\cdot c^T(V_{\leq 0}-S^{\prime }_0)$ .

Using the isomorphism $H_T^*\mathrm {Gr}(V)=H_T^*\mathrm {Gr}(\mathbb {V})$ , the homomorphism $\boxplus ^*=\Delta $ determines a commutative coproduct structure on $H_T^*\mathrm {Gr}(V)$ . This coproduct has been studied by many authors. It is induced by the coproduct on $\Lambda $ , and it is well known that this can be written in the Schur basis by

$$\begin{align*}\Delta(s_\lambda(\mathbf{c})) = \sum_{\mu,\nu} c_{\mu,\nu}^\lambda s_{\mu}(c) \otimes s_{\nu}(c'), \end{align*}$$

where $c_{\mu ,\nu }^\lambda $ is the Littlewood-Richardson coefficient. So it can be computed from an expression in terms of the Schur basis. (Using (4), the Schur function $s_\lambda (c)$ is defined as the determinant

$$ \begin{align*} s_\lambda(c) = s_\lambda(c|0) = \det( c_{\lambda_i-i+k} )_{1\leq i,j \leq s} \end{align*} $$

for any partition $\lambda _1\geq \cdots \geq \lambda _s \geq 0$ .)

We are more interested in the Schubert basis. Schubert varieties in $\mathrm {Gr}(\mathbb {V})$ are defined with respect to a flag $\mathbb {V}_ {\scriptscriptstyle {\bullet }} ^-$ , where for $q\in \mathbb {Z}$ ,

(7) $$ \begin{align} \mathbb{V}^-_q = V_{>0} \oplus V_{>q}. \end{align} $$

Then $\mathbf {\Omega }_\lambda = \{ \mathbb {E}\,|\, \dim (\mathbb {E} \cap \mathbb {V}^-_{\lambda _i-i})\geq i\text { for all }i \}$ . Under the embedding $f\colon \mathrm {Gr}(V) \to \mathrm {Gr}(\mathbb {V})$ , we have $f^{-1}\mathbf {\Omega }_\lambda = \Omega _\lambda $ , so $f^*[\mathbf {\Omega }_\lambda ] = [\Omega _\lambda ]$ and

(8) $$ \begin{align} [\mathbf{\Omega}_\lambda] = \mathbf{S}_{w_\lambda}(\mathbf{c};x;y) = s_\lambda(\mathbf{c}|y). \end{align} $$

(To see $f^{-1}\mathbf {\Omega }_\lambda = \Omega _\lambda $ , note that $(V_{\leq 0}\oplus E) \cap \mathbb {V}^-_{\lambda _i-i} = (V_{\leq 0} \oplus E) \cap (V_{>0} \oplus V_{>\lambda _i-i}) = 0 \oplus (E \cap V_{>\lambda _i-i})$ , so the equations defining $\mathbf {\Omega }_\lambda $ pull back to those defining $\Omega _\lambda $ . The formula $f^*[\mathbf {\Omega }_\lambda ] = [\Omega _\lambda ]$ , and implies (8), although the latter can also be proved directly.)

Molev gives formulas for the structure constants here [Reference Molev24]. In our geometric context, we have

$$\begin{align*}\boxplus^*[\mathbf{\Omega}_\lambda] = \sum_{\mu,\nu} \widehat{c}_{\mu,\nu}^\lambda(y) \, [\Omega_\mu] \times [\Omega_\nu], \end{align*}$$

for dual Littlewood-Richardson polynomials $\widehat {c}_{\mu ,\nu }^\lambda (y) \in \mathbb {Z}[y]$ . In terms of Schubert polynomials, this is equivalent to the Cauchy formula:

$$ \begin{align*} \mathbf{S}_{w_\lambda}(\mathbf{c};x;y) &= \sum_{uv \dot{=} w_\lambda} F_{u}(c;y) \cdot \mathbf{S}_{v}(c';x;y) \\ &= \sum_{\mu,\nu \subset \lambda} \widehat{c}_{\mu,\nu}^\lambda(y)\,\mathbf{S}_{w_\mu}(c;x;y)\cdot \mathbf{S}_{w_{\nu}}(c';x;y). \end{align*} $$

(See [Reference Anderson and Fulton5, §5] and [Reference Lam, Lee and Shimozono21, §4.8].Footnote ² ) That is, for $u=w_\lambda w_\nu ^{-1}$ , the Stanley function expands as $F_u(c;y) =\sum _{\mu } \widehat {c}_{\mu ,\nu }^\lambda (y)\,\mathbf {S}_{w_\mu }(c;x;y)$ . The polynomial $\mathbf {S}_{w_\lambda }(c;x;y)=s_\lambda (c|y)$ is always independent of x since it represents a class coming from $H_T^*\mathrm {Gr} = \Lambda [y]$ .

The coefficients $\widehat {c}_{\mu ,\nu }^\lambda (y)$ are Graham-positive; this is a special case of [Reference Lam, Lee and Shimozono21, Theorem 4.22]. We will give an argument which establishes the general case (and also applies to this case) when proving Theorem 8.5 below.

Example 8.4. The nonzero coefficients for $\lambda =(3,1)$ are

$$ \begin{align*} &\widehat{c}^{(3,1)}_{\emptyset,(3,1)} = \widehat{c}^{(3,1)}_{(1),(2,1)} = \widehat{c}^{(3,1)}_{(1),(3)} = \widehat{c}^{(3,1)}_{(2),(1,1)} = \widehat{c}^{(3,1)}_{(2),(2)} = 1, \\ &\widehat{c}^{(3,1)}_{(1),(2)} = y_0-y_1, \\ & \widehat{c}^{(3,1)}_{(1),(1,1)} = y_2-y_1, \\ & \widehat{c}^{(3,1)}_{(1),(1)} = (y_2-y_1)(y_0-y_1). \end{align*} $$

One can have repeated factors – for example, $\widehat {c}^{(2,2,1)}_{(1),(1,1)} = (y_0-y_1)^2$ . In fact, we will see that only linear forms and squares of linear forms occur as factors (Theorem 8.5).

As usual, the morphism $\boxplus $ comes from compatible morphisms of finite-dimensional varieties, $\boxplus \colon {Gr} (m,V_{(-m,m]}) \times {Gr} (m,V_{(-m,m]}) \to {Gr} (2m,\mathbb {V}_{(-m,m]})$ . The subvariety $\boxplus ( X_\mu \times X_\nu ) \subseteq {Gr} (2m,\mathbb {V}_{(-m,m]})$ is a Richardson variety, $X_{\mu \oslash _m \nu } \cap \Omega _{\rho _m}$ , where $\mu \oslash _m \nu $ is the partition $(\nu _1+m,\ldots ,\nu _m+m,\mu _1,\ldots ,\mu _m)$ , and $\rho _m$ is the $m\times m$ rectangle. (In Young diagrams, one forms $\mu \oslash _m \nu $ by placing $\nu $ to the right of the $m\times m$ rectangle and placing $\mu $ below the rectangle; we are assuming m is at least equal to the number of parts of $\nu $ and to the largest part of $\mu $ . This is [Reference Thomas and Yong27, Proposition 2.1].) The coefficients $\widehat {c}_{\mu ,\nu }^\lambda $ arise in the expansion of the class of this Richardson variety in a Schubert basis with respect to a third T-invariant flag: the one corresponding to the ordered basis

$$ \begin{align*} &(e_{-m+1},0), \ldots, (e_0,0), (0,e_{-m+1}), \ldots, (0,e_0), \\ & \qquad \qquad \qquad (0,e_1),\ldots,(0,e_m), (e_1,0), \ldots, (e_m,0). \end{align*} $$

This interpretation leads to another way of computing. Fix a sufficiently large m, consider variable sets $x=(x_{-2m+1},\ldots ,x_{2m})$ and $t=(t_{-2m+1},\ldots ,t_{2m})$ , and let $s_\lambda (c|t)$ be the specialization of $\mathbf {S}_{w_\lambda }(c;x;t)$ by $c = \prod _{i=-2m+1}^{0} \frac {1+t_i}{1-x_i}$ . Then $\widehat {c}_{\mu ,\nu }^\lambda (y)$ is the coefficient of $s_{\mu \oslash _m \nu }(c|{\mathbf {y}})$ in the expansion of $s_\lambda (c|\widetilde {\mathbf {y}} ) \cdot s_{\rho _m}(c|{\mathbf {y}})$ , where

(9) $$ \begin{align} \mathbf{y} &= (y_{-m+1},\ldots,y_m,y_{-m+1},\ldots,y_m) \end{align} $$

and

(10) $$ \begin{align} \widetilde{\mathbf{y}} &= (y_{-m+1},\ldots,y_0,y_{-m+1},\ldots,y_0,y_1,\ldots,y_m,y_1,\ldots,y_m). \end{align} $$

For example, $\widehat {c}^{(2,2,1)}_{(1),(1,1)}(y) = (y_0-y_1)^2$ is the coefficient of $s_{(3,3,1)}(c|{\mathbf {y}})$ in the product

$$\begin{align*}s_{(2,2,1)}(c| y_{-1},y_0,y_{-1},y_0,y_1,y_2,y_1,y_2) \cdot s_{(2,2)}(c|y_{-1},y_0,y_1,y_2,y_{-1},y_0,y_1,y_2). \end{align*}$$

(In comparison with [Reference Lam, Lee and Shimozono21], our $s_\lambda (c|t)$ is their $s_\lambda (x|\!|{-a})$ .)

8.2 Flag varieties

The direct sum morphism extends to an action on the flag variety: one defines

$$\begin{align*}\boxplus\colon \mathrm{Gr}(V)\times \mathrm{Fl}(V) \to \mathrm{Fl}(\mathbb{V}) \end{align*}$$

in the same way, so that $(F,E_\bullet )$ is sent to the flag $\mathbb {E}_\bullet $ with $\mathbb {E}_k = F\oplus E_k$ . The pullback $\boxplus ^*\colon H_T^*\mathrm {Fl} \to H_T^*(\mathrm {Gr}\times \mathrm {Fl})$ is identified with a co-module operation $\Delta \colon \Lambda [x;y] \to \Lambda [y] \otimes _{\mathbb {Z}[y]} \Lambda [x;y]$ . As before, this homomorphism is determined by its values on Schur polynomials, and one can compute using classical Littlewood-Richardson numbers; but also as before, we are more interested in the behavior of Schubert polynomials.

The morphism $\boxplus $ induces an embedding $f\colon \mathrm {Fl}(V) \hookrightarrow \mathrm {Fl}(\mathbb {V})$ , by $E_\bullet \mapsto V_{\leq 0}\oplus E_\bullet $ , and as in Proposition 8.1, the pullback is an isomorphism on cohomology rings, $\Lambda [x,y] \mapsto \Lambda [x,y]$ , sending $\mathbf {c}\to c$ .

Schubert varieties in $\mathrm {Fl}(\mathbb {V})$ are again defined with respect to the flag $\mathbb {V}^-_\bullet $ described in (7), so

$$\begin{align*}\mathbf{\Omega}_w = \big\{ E_\bullet \,|\, \dim(E_p \cap \mathbb{V}^-_q) \geq k_w(p,q) \text{ for all }p,q \big\}. \end{align*}$$

As before, $f^{-1}\mathbf {\Omega }_w = \Omega _w$ , and we have $[\mathbf {\Omega }_w] = \mathbf {S}_w(\mathbf {c};x;y)$ in $H_T^*\mathrm {Fl}(\mathbb {V})$ .

The action on Schubert classes is by

$$\begin{align*}\boxplus^*[\mathbf{\Omega}_w] = \sum_{\mu,v} \widehat{c}^w_{\mu,v}(y) [\Omega_\mu] \times [\Omega_v]. \end{align*}$$

Using $\mathbf {c}=c\cdot c'$ , this is expressed via the Cauchy formula as

$$ \begin{align*} \mathbf{S}_w(\mathbf{c};x;y) &= \sum_{uv \dot{=} w} F_{u}(c;y) \cdot \mathbf{S}_{v}(c';x;y) \\ &= \sum_{\mu , v} \widehat{c}_{\mu,v}^w(y)\,\mathbf{S}_{w_\mu}(c;x;y)\cdot \mathbf{S}_{v}(c';x;y). \end{align*} $$

Comparing coefficients of $\mathbf {S}_v$ , it follows that $\widehat {c}_{\mu ,v}^w(y)=0$ unless $\ell (wv^{-1})=\ell (w)-\ell (v)$ . When this length-additivity condition holds, the coefficients arise in the expansion

$$\begin{align*}F_{wv^{-1}}(c;y) =\sum_{\mu} \widehat{c}_{\mu,v}^w(y)\,\mathbf{S}_{w_\mu}(c;x;y). \end{align*}$$

In the terminology of [Reference Lam, Lee and Shimozono21, §4], these are the double Edelman-Greene coefficients, the precise translation being

$$\begin{align*}\widehat{c}^w_{\mu,v}(y) = \widehat{c}_{\mu,e}^{wv^{-1}}(y) = j^{wv^{-1}}_\mu(-a) \end{align*}$$

when $\ell (wv^{-1})=\ell (w)-\ell (v)$ (and $\widehat {c}^w_{\mu ,v}(y) = 0$ otherwise).

The total order $\prec $ on $\mathbb {Z}$ is the one defined in Proposition 8.3, so $1\prec 2 \prec \cdots \prec -1 \prec 0$ . The theorem refines [Reference Lam, Lee and Shimozono21, Theorem 4.22], which asserts positivity without bounds on the powers of $y_i-y_j$ . The proof given in [Reference Lam, Lee and Shimozono21] relates the coefficient $\widehat {c}_{\mu ,v}^w(y)$ to one appearing in the equivariant homology of the affine Grassmannian, and then invokes the quantum-affine (Peterson) isomorphism and positivity in equivariant quantum cohomology.

Our argument is based on a direct application of Graham’s positivity theorem [Reference Graham15], which says the following. Suppose $B_N$ is a connected solvable group, with unipotent radical $U_N$ and maximal torus T, and $B_0\subset B_N$ is a closed subgroup whose unipotent radical $U_0\subset U_N$ is normalized by T. Let $\chi _1,\ldots ,\chi _N$ be the characters of T on the quotient variety $U_N/U_0$ (considered as an affine space), counted with multiplicity. If $B_N$ acts on a variety X, and $Y \subseteq X$ is a $B_0$ -invariant subvariety, then there are $B_N$ -invariant cycles $Z_I$ so that

$$\begin{align*}[Y] = \sum_{I\subseteq \{1,\ldots,N\}} \left(\prod_{i\in I} \chi_i \right) [Z_I] \end{align*}$$

as T-equivariant Chow (or homology) classes. (See also [Reference Anderson and Fulton6, Ch. 19].)

Proof. We may compute a given coefficient $c^w_{\mu ,v}$ on a sufficiently large but finite dimensional flag variety, so for now, we choose $m\gg 0$ and set $V=V_{(-m,m]}$ , etc., writing $ {Fl} (V)$ for the complete flag variety, and $ {Fl} (\mathbb {V}) = {Fl} (m,m+1,\ldots ,3m;\mathbb {V})$ for the partial flag variety, so the direct sum map is $\boxplus \colon {Gr} (m,V)\times {Fl} (V) \to {Fl} (\mathbb {V})$ . We use the ordered basis $e_{-m+1}, \ldots , e_m$ for V, as usual, and let $B^+\subseteq GL(V)$ be the subgroup stabilizing the corresponding flag $V_{\leq \bullet }$ . For $\mathbb {V}= V\oplus V$ , we use the ordered basis

$$ \begin{align*} &(e_{-m+1},0), \ldots, (e_0,0), (0,e_{-m+1}), \ldots, (0,e_0), \\ & \qquad \qquad \qquad (0,e_1),\ldots,(0,e_m), (e_1,0), \ldots, (e_m,0). \end{align*} $$

The flag $\mathbb {V}^-_\bullet $ obtained by reading this basis backwards is the one used to define the (opposite) Schubert variety $\mathbf {\Omega }_w$ . Let $\mathbb {B}^-\subseteq GL(\mathbb {V})$ be the subgroup stabilizing this flag, and let $\mathbb {B}^+$ be the subgroup stabilizing the flag $\mathbb {V}^+_\bullet $ obtained by reading the basis forwards.

So in our chosen bases for V and $\mathbb {V}$ , the subgroups $B^+\subset GL(V)$ and $\mathbb {B}^+\subset GL(\mathbb {V})$ are upper-triangular, and $B^-$ and $\mathbb {B}^-$ are lower-triangular. Let $U^+\subset B^+$ and $\mathbb {U}^+\subset \mathbb {B}^+$ be the corresponding unipotent radicals.

In $ {Fl} (V)$ , the $B^-$ invariant Schubert varieties $\Omega _v$ (of codimension $\ell (v)$ ) are transverse to $B^+$ -invariant Schubert varieties $X_v$ (of dimension $\ell (v)$ ); likewise one has $\Omega _\mu $ and $X_\mu $ in $ {Gr} (m,V)$ . The $\mathbb {B}^-$ -invariant $\mathbf {\Omega }_w$ and $\mathbb {B}^+$ -invariant $\mathbf {X}_w$ in $ {Fl} (\mathbb {V})$ are defined with respect to the flags $\mathbb {V}^-_\bullet $ and $\mathbb {V}^+_\bullet $ , respectively. As we have seen, $\mathbf {\Omega }_w$ has class $\mathbf {S}_w(\mathbf {c};x;y)$ .

By Poincaré duality, we have

$$\begin{align*}\boxplus_*( [X_\mu \times X_v ] ) = \sum_w \widehat{c}^w_{\mu,v}(y) \cdot [\mathbf{X}_w] \end{align*}$$

in $H_T^* {Fl} (\mathbb {V})$ . The left-hand side is the class of the $(B^+ \times B^+)$ -invariant subvariety $\boxplus ( X_\mu \times X_v ) \subseteq {Fl} (\mathbb {V})$ . Applying Graham’s theorem expresses this as a sum of $\mathbb {B}^+$ -invariant cycles, with coefficients coming from the characters of T acting on $\mathbb {U}^+/(U^+\times U^+)$ . Since the only $\mathbb {B}^+$ -invariant cycles are Schubert varieties $\mathbf {X}_w$ , this is the desired decomposition.

The characters on $\mathbb {U}^+/(U^+\times U^+)$ are $y_i-y_j$ for $i\leq 0$ and $j>0$ (each with multiplicity $2$ ), and $y_i-y_j$ for $i,j\leq 0$ or $i,j>0$ (each with multiplicity $1$ ). See Figure 2 for an illustration.

Figure 2

Weights ( $\bullet $ ) on $U^+\times U^+$ and ( $\circ $ ) on $\mathbb {U}^+/(U^+\times U^+)$ .

At this point, we have established that $\widehat {c}^w_{\mu ,v}(y)$ is a nonnegative sum of monomials in $y_i-y_j$ for $i\leq 0$ and $j>0$ (each occurring at most twice) and $y_i-y_j$ for $i,j$ of the same sign (each occurring at most once). To conclude, observe that if i and j have the same sign and $i<j$ , the linear forms $y_i-y_j$ cannot contribute since this would violate [Reference Lam, Lee and Shimozono21, Theorem 4.22].

In fact, the direct sum morphism is equivariant for a larger torus. Let $\mathbb {T}=T\times T'$ act on $\mathbb {V} = V\oplus V$ by characters y on the first factor and $y'$ on the second factor. Then $\boxplus \colon \mathrm {Gr}(V) \times \mathrm {Fl}(V) \to \mathrm {Fl}(\mathbb {V})$ is equivariant for the induced $\mathbb {T}$ -action. One can define coefficients $\widehat {c}_{\mu ,v}^w(y,y') \in \mathbb {Z}[y;y']$ by

$$\begin{align*}\boxplus^*[\mathbf{\Omega}_w] = \sum_{\mu,v} \widehat{c}_{\mu,v}^w(y,y')\, [\Omega_\mu] \times [\Omega_v], \end{align*}$$

or equivalently,

$$\begin{align*}\boxplus_*[X_\mu \times X_v] = \sum_{\mu,v} \widehat{c}_{\mu,v}^w(y,y')\, [\mathbf{X}_w]. \end{align*}$$

The argument for Theorem 8.5 also proves that these coefficients are also Graham-positive:

Theorem 8.7. The coefficient $\widehat {c}^w_{\mu ,v}(y,y')$ is a nonnegative sum of squarefree monomials in linear forms

$$\begin{align*}y_- -y^{\prime}_-,\; y_- - y^{\prime}_+,\; y^{\prime}_- - y_+,\; \text{ and }y^{\prime}_+-y_+, \end{align*}$$

where $y_+$ stands for any $y_i$ with $i>0$ , $y_-$ for $y_i$ with $i\leq 0$ , etc.

In other words, the forms appearing are $d-c$ with $c\prec d$ , where c and d are among the y and $y'$ variables, ordered so that

$$\begin{align*}\{y_+\} \prec \{ y^{\prime}_+\} \prec \{ y^{\prime}_- \} \prec \{ y_-\} , \end{align*}$$

and exactly one of c or d is a primed variable. (To compare with the illustration in Figure 2, label the rows and columns by $-1,0,-1',0',1',2',1,2$ , so that they are scaled by the corresponding characters $y_i$ and $y^{\prime }_i$ .)

The coefficients are equal to the triple Edelman-Greene coefficients $j^w_\mu (a,b)$ of [Reference Lam, Lee and Shimozono21, §10], after setting $y_i=-b_i$ and $y^{\prime }_i=-a_i$ ; that is, $j^{w}_\mu (a,b) = \widehat {c}_{\mu ,e}^w(-b,-a)$ . Indeed, the definition shows that $\widehat {c}^w_{\mu ,v}(y,y')$ are the coefficients appearing in the expansion

$$\begin{align*}\mathbf{S}_w(\mathbf{c};x;y') = \sum_{\mu,v} \widehat{c}^w_{\mu,v}(y,y')\, s_\mu(c|y)\,\mathbf{S}_v(c';x;y'), \end{align*}$$

which, noting our sign conventions, agrees with the characterization of $j^{wv^{-1}}_\mu (a,b)$ from [Reference Lam, Lee and Shimozono21, §10]. So the theorem expresses positivity in the a and b variables, answering a question raised in [Reference Lam, Lee and Shimozono21, Remark 10.13].

One recovers the coefficients $\widehat {c}^w_{\mu ,v}(y)$ by setting $y'=y$ . However, Theorem 8.5 does not follow from Theorem 8.7 since one can see factors of $y^{\prime }_i-y_j$ with $i\prec j$ .

Example 8.8. We have

$$ \begin{align*} \widehat{c}^{[2,3,-1,0,1]}_{(2,2),e}(y,y') &= (y^{\prime}_1-y_2)(y^{\prime}_1-y_1) \end{align*} $$

and

$$ \begin{align*} \widehat{c}^{[2,3,-1,0,1]}_{(1,1),\, [0,2,-1,1]}(y,y') &= y^{\prime}_1-y_1. \end{align*} $$

This shows there is no total order $\prec $ on the variables $(y,y')$ such that both (1) the coefficients $\widehat {c}^w_{\mu ,v}(y,y')$ are nonnegative sums of monomials in $d-c$ , with $c\prec d$ , and (2) the specialization $y'=y$ respects the order (i.e., $y_i \prec y^{\prime }_j$ implies $y_i\prec y_j$ ). (Of course, any coefficient $\widehat {c}^w_{\mu ,v}(y,y')$ violating (2) must map to $0$ under the specialization $y'=y$ , as the two shown above do.)

Consider the corresponding direct sum morphism $\boxplus \colon {Gr} (m,V_{(-m,m]}) \times {Fl} (V_{(-m,m]}) \to {Fl} (m,m+1,\ldots ,3m;\mathbb {V}_{(-m,m]})$ of finite-dimensional varieties, and identify $\mathbb {V}_{(-m,m]} = V_{(-2m,2m]}$ using the ordered basis which lists $(e_{i},0)$ , and then $(0,e_{i})$ . As before, the image of $X_\mu \times X_v$ under direct sum is a Richardson variety. Specifically, define a permutation of $\{-2m+1,\ldots ,2m\}$ by

$$ \begin{align*} \mu \oslash_m v &= [w_\mu(-m+1)-m,\ldots,\,w_\mu(0)-m,\,v(-m+1)+m,\ldots \\ & \qquad \ldots,\, v(m)+m,\,w_\mu(1)-m,\ldots,\,w_\mu(m)-m ]. \end{align*} $$

For example, for $\mu =(3,1,1)$ , $v=[0,-1,2,-2,3,1]$ , and $m=3$ , we have $\mu \oslash _m v = [-4,-3,0,3,2,5,1,6,4,-5,-2,-1]$ .

Proposition 8.10. Assume m is large enough so that $w_\mu $ and v lie in $\mathcal {S}_{(-m,m]}$ . Let $x^{(m)} = [-2m+1,\ldots ,\,-m,\,1,\ldots ,\,2m,\,-m+1,\ldots ,\,0]$ . Then

$$\begin{align*}\boxplus( X_\mu \times X_v ) = X_{\mu \oslash_m v} \cap \Omega_{x^{(m)}}, \end{align*}$$

a Richardson variety in $ {Fl} (m,m+1,\ldots ,3m;\mathbb {V}_{(-m,m]})$ .

The proof is the same as that of [Reference Thomas and Yong27, Proposition 2.1]. This leads to another way of computing the Edelman-Greene coefficients.

Corollary 8.11. The polynomial $\widehat {c}_{\mu ,v}^{w}(y,y')$ is equal to the coefficient of $\mathbf {S}_{\mu \oslash _m v}(\mathbf {c};x;\mathbf {y})$ in the expansion of $\mathbf {S}_{w}(\widetilde {\mathbf {c}};x;\widetilde {\mathbf {y}})\cdot \mathbf {S}_{x^{(m)}}(\mathbf {c};x;\mathbf {y})$ , where

$$ \begin{align*} \mathbf{y} &= (y_{-m+1},\ldots,y_m,y^{\prime}_{-m+1},\ldots,y^{\prime}_m) \end{align*} $$

and

$$ \begin{align*} \widetilde{\mathbf{y}} &= (y_{-m+1},\ldots,y_0,y^{\prime}_{-m+1},\ldots,y^{\prime}_m,y_1,\ldots,y_m), \end{align*} $$

and $\mathbf {c}$ and $\widetilde {\mathbf {c}}$ are determined by specializing $\prod _{i=-2m+1}^0\frac {1+t_i}{1-x_i}$ to $t=\mathbf {y}$ and $t=\widetilde {\mathbf {y}}$ , respectively.

Proof. The specializations of the y variables ensure that $\mathbf {S}_w(\widetilde {\mathbf {c}};x;\widetilde {\mathbf {y}}) = [\mathbf {\Omega }_w]$ , $\mathbf {S}_{\mu \oslash _m v}(\mathbf {c};x;\mathbf {y}) = [\Omega _{\mu \oslash _m v}]$ , and $\mathbf {S}_{x^{(m)}}(\mathbf {c};x;\mathbf {y}) = [\Omega _{x^{(m)}}]$ in $H_T^* {Fl} (m,m+1,\ldots ,3m;\mathbb {V}_{(-m,m]})$ . And by Poincaré duality, the coefficient of $[\Omega _{\mu \oslash _m v}]$ in the expansion of $[\mathbf {\Omega }_w]\cdot [\Omega _{x^{(m)}}]$ is equal to the (equivariant) integral

$$\begin{align*}\int_{ {Fl} (\mathbb{V})} [\mathbf{\Omega}_w]\cdot[\Omega_{x^{(m)}}]\cdot [X_{\mu\oslash_m v}]. \end{align*}$$

We have $[\Omega _{x^{(m)}}]\cdot [X_{\mu \oslash _m v}] = [ \Omega _{x^{(m)}} \cap X_{w_\mu \oslash _m v} ] = [ \boxplus ( X_\mu \times X_v )]$ , so this integral becomes

$$\begin{align*}\int_{ {Fl} (\mathbb{V})} [\mathbf{\Omega}_w]\cdot \boxplus_*[X_\mu \times X_v ] = \int_{ {Gr} (V)\times {Fl} (V)} \boxplus^*[\mathbf{\Omega}_w] \cdot [X_\mu \times X_v ], \end{align*}$$

which is the coefficient of $[\Omega _\mu ]\times [\Omega _v]$ in $\boxplus ^*[\mathbf {\Omega }_w]$ , as claimed.

9.1 Lagrangian Grassmannians and isotropic flag varieties

We fix a standard symplectic form on V, defined by setting

$$\begin{align*}\langle e_{1-i}, e_i \rangle = -\langle e_i, e_{1-i} \rangle = 1 \end{align*}$$

for $i>0$ , and setting all other pairings to $0$ . The form

$$\begin{align*}\langle \;,\; \rangle \colon V \otimes V \to \mathbb{C}_z \end{align*}$$

is preserved by $\mathbf {T}$ , where the target $\mathbb {C}_z$ is scaled by character z. When restricted to each $2m$ -dimensional subspace $V_{(-m,m]}$ , this defines a symplectic form and an isomorphism

$$\begin{align*}V_{(-m,m]} \xrightarrow{\sim} V_{(-m,m]}^*\otimes \mathbb{C}_z. \end{align*}$$

Using these subspaces to define the restricted dual of V, this also gives an isomorphism $V \xrightarrow {\sim } V^{*'}\otimes \mathbb {C}_z$ .

We fix the flag $V_{\leq \bullet }$ as before. The infinite Lagrangian Grassmannian is the subvariety

$$\begin{align*}\mathrm{LG} \subseteq \mathrm{Gr} \end{align*}$$

parametrizing subspaces $E\subseteq V$ which belong to $\mathrm {Gr}$ and are isotropic with respect to the symplectic form (i.e., those E for which $\langle \;,\;\rangle $ becomes identically zero when restricted to E). As for $\mathrm {Gr}$ , we use the notation $\mathrm {LG}(V;V_{\leq \bullet })$ when there is ambiguity in the flag.

The subspace $V_{\leq 0}$ is isotropic, so it lies in $\mathrm {LG}$ . The subspace $V_{>0}$ is also isotropic, but it does not lie in $\mathrm {Gr}$ so does not define a point of $\mathrm {LG}$ . (Note, however, that the symplectic form defines isomorphisms $V_{\leq 0} \cong V_{>0}^{*'} \otimes \mathbb {C}_z$ .)

As noted in the introduction, one has compatible embeddings

making $\mathrm {LG} = \bigcup _{m>0}LG(m,V_{(-m,m]})$ .

The cohomology ring of each finite-dimensional Lagrangian is generated by Chern classes of the tautological bundle $S \subseteq V_{(-m,m]}$ , with relations coming from the Whitney sum formula. Using $c=c^T(V_{\leq 0}-S)$ , these relations are determined by $c\cdot \overline {c} = 1$ , where

$$\begin{align*}\overline{c} = c^T(V_{\leq0}^*\otimes \mathbb{C}_z - S^*\otimes \mathbb{C}_z). \end{align*}$$

(Using the symplectic form, one has $V_{(-m,m]}/S \cong S^*\otimes \mathbb {C}_z$ and $V_{\leq 0}^*\otimes \mathbb {C}_z = V_{>0}$ , so the relations follow.) By standard Chern class identities, one writes

$$\begin{align*}\overline{c}_p = \sum_{i=1}^p \binom{p-1}{i-1}\, (-z)^{p-i} (-1)^i\, c_i. \end{align*}$$

Extracting the degree $2p$ part of $c\cdot \overline {c}$ , one finds relations

$$\begin{align*}C_{pp} := \sum_{0\leq i\leq j \leq p} (-1)^j \left( \binom{j}{i}+\binom{j-1}{i} \right) z^i\, c_{p-i+j}\, c_{p-j} = 0, \end{align*}$$

for $p>0$ . Taking the limit, we have

$$\begin{align*}H_{\mathbf{T}}^*\mathrm{LG} = \mathbf{\Gamma}[y_+], \end{align*}$$

where

$$\begin{align*}\mathbf{\Gamma} = \Lambda[z]/( C_{pp} )_{p>0}. \end{align*}$$

Pullback by the inclusion $\mathrm {LG} \hookrightarrow \mathrm {Gr}$ induces the canonical surjection $\Lambda [z][y] \twoheadrightarrow \mathbf {\Gamma }[y_+]$ .

For $k\leq 0$ , one defines $\mathrm {IG}^k \subseteq \mathrm {Gr}^k$ in the same way. It is the union

$$\begin{align*}\mathrm{IG}^k = \bigcup_{m>|k|} IG(m+k,V_{(-m,m]}) \end{align*}$$

of (possibly non-maximal) isotropic Grassmannians. The (type C) infinite isotropic flag variety is the variety

$$\begin{align*}\mathrm{Fl}^C = \{ E_\bullet : (\cdots \subset E_{-1} \subset E_{0} =E \subset V) \,|\, E_i \in \mathrm{IG}^{i} \}, \end{align*}$$

a subvariety of $\prod _{k\leq 0} \mathrm {IG}^k$ . Its cohomology ring is

$$\begin{align*}H_{\mathbf{T}}^*\mathrm{Fl}^C = \mathbf{\Gamma}[x_+,y_+], \end{align*}$$

using $x_i = c_1^{\mathbf {T}}(S_{-i+1}/S_{-i})$ for $i>0$ , where $(\cdots \subset S_{-1} \subset S_{0} =S \subset V)$ is the tautological flag. (As usual, these should be regarded as the stable limits of vector bundles on the finite-dimensional type C flag varieties.)

Just as for finite-dimensional varieties, an isotropic flag extends canonically to a complete flag, by $E_{i} = E_{-i}^\perp $ for $i>0$ , and one obtains an embedding $\mathrm {Fl}^C \hookrightarrow \mathrm {Fl}$ . Using the symplectic form to identify $V\cong V^{*'}\otimes \mathbb {C}_z$ , this realizes $\mathrm {Fl}^C$ as the fixed locus for the duality involution described in §7.1 (or rather, a variation of that involution which twists by $\mathbb {C}_z$ ; see [Reference Anderson and Fulton5]). In particular, we have $E_{i}/E_{i-1} \cong (E_{1-i}/E_{-i})^*\otimes \mathbb {C}_z$ for $i\geq 1$ .

The pullback on cohomology is the surjection $\Lambda [z][x,y] \twoheadrightarrow \mathbf {\Gamma }[x_+,y_+]$ , where $x_i \mapsto x_i$ for $i>0$ , and $x_i\mapsto z-x_{1-i}$ for $i\leq 0$ . Realizing $\mathrm {Fl}^C\subset \mathrm {Fl}$ as the fixed locus of a (twisted) duality involution gives another way of viewing the relations defining this quotient of $\Lambda [z][x,y]$ . The corresponding homomorphism

$$\begin{align*}\boldsymbol\omega(c_k) = \sum_{i=1}^k \binom{k-1}{i-1} (-z)^{k-i} S_{1^i}(c), \quad \boldsymbol\omega(x_i)=z-x_{-i}, \quad \boldsymbol\omega(y_i) = z-y_{-i} \end{align*}$$

must be the identity on $H_{\mathbf {T}}^*\mathrm {Fl}^C$ , and the relations express this.

Remark 9.2. In symmetric function theory, one often embeds $\Gamma \hookrightarrow \Lambda $ , considering both as rings of symmetric functions in auxiliary variables. The ring $\mathbf {\Gamma }$ also embeds in $\Lambda [z]$ . This requires more care, but it also points the way to a geometric interpretation. It is helpful to realize these inclusions of rings as pullbacks via a different map between infinite Grassmannians. We will describe it in terms of compatible maps of finite-dimensional varieties.

To lighten the notation, let $V_m = V_{(-m,m]}$ and $L=\mathbb {C}_z$ , and let $\mathbb {V}_m = V_m \oplus V^{*}_m \otimes L$ , with its canonical L-valued symplectic form. For any fixed k, there is a map

$$\begin{align*}{Gr} (m+k,V_m) \hookrightarrow LG(\mathbb{V}_m), \end{align*}$$

sending a point $A\subset V_m$ to $A\oplus (V_m/A)^*\otimes L \subset \mathbb {V}$ . One checks that this is an isotropic subspace. The space $\mathbb {E}_m = V_{\leq 0} \oplus V_{>0}^* \otimes L \subset \mathbb {V}_m$ is also isotropic subspace. Let $\mathbb {S}\subset \mathbb {V}_m$ be the tautological bundle. Pullback sends $c^{\mathbf {T}}(\mathbb {V}_m - \mathbb {S} - \mathbb {E}_m)$ to

$$\begin{align*}c^{\mathbf{T}}(\mathbb{V}_m - S - Q^*\otimes L - \mathbb{E}_m ) = c^{\mathbf{T}}( V_{>0}-V_{>0}^*\otimes L + S^*\otimes L - S), \end{align*}$$

where $S \subset V_m \twoheadrightarrow Q$ are tautological bundles on $ {Gr} (m+k,V_m)$ .

These maps are all compatible with the natural inclusions $V_m\subset V_{m+1}$ . So there is a corresponding morphism $\mathrm {Gr}^{(k)}(V) \to \mathrm {LG}(\mathbb {V})$ . The corresponding pullback map on cohomology, $\mathbf {\Gamma } \to \Lambda [y_+][z]$ is given by

(11) $$ \begin{align} c \mapsto \prod_{i> 0} \frac{1+y_i}{1-y_i+z} \prod_{i\leq k} \frac{1+x_i+z}{1-x_i}, \end{align} $$

where $x_{-m+1},\ldots ,x_k$ are Chern roots of $S^*$ on each finite-dimensional $ {Gr} (m+k,V_m)$ , and $\Lambda $ is regarded as the ring of supersymmetric functions in the variables $x_i$ for $i\leq k$ and $y_i$ for $i>0$ . The series on the right-hand side of (11) is stable with respect to setting $x_i=y_i=0$ for $|i|>m$ , so its homogeneous pieces are well-defined elements of $\Lambda [y_+][z]$ , as they must be. (They are deformations of the classical polynomials $Q_p(x)$ .)

9.2 Schubert varieties and Schubert polynomials

The group of signed permutations is the subgroup $W_\infty \subset \mathcal {S}_{\mathbb {Z}}$ of permutations w such that $w(1-i)=1-w(i)$ for all i. These are the elements of $\mathcal {S}_{\mathbb {Z}}$ which are fixed by the involution $\omega $ . The submonoid $\mathrm {SgnInj}(\mathbb {Z}) \subset \operatorname {\mathrm {Inj}}(\mathbb {Z})$ is defined similarly, and one also has the submonoid $\mathrm {SgnInj}^0(\mathbb {Z}) \subset \mathrm {SgnInj}(\mathbb {Z})$ of signed injections with finitely many sign changes. (The balancing condition is automatic here.) Choosing a large enough m so that $w(i)=i$ for $|i|>m$ , we often write $w\in W_\infty $ in one-line notation as $w=[w(1),\ldots ,w(m)]$ .

Just as $\operatorname {\mathrm {Inj}}^0(\mathbb {Z})$ indexes fixed points of $\mathrm {Fl}$ , the subset $\mathrm {SgnInj}^0(\mathbb {Z})$ indexes fixed points of $\mathrm {Fl}^C$ : the point $p_w$ corresponds to the flag $E_\bullet $ with $E_k$ spanned by $e_{w(i)}$ for $i\leq 0$ . (With conventions as in §6 for integers not in the image of w.)

Schubert varieties are indexed by signed permutations. For each $w \in W_\infty $ , there is a Schubert variety in $\mathrm {Fl}^C$ , defined by

$$\begin{align*}\Omega_w = \{ E_\bullet \,|\, \dim(E_p \cap V_{>q}) \geq k_w(p,q) \text{ for } p\leq 0 \text{ and all }q \}, \end{align*}$$

where $k_w(p,q) = \#\{ a\leq p \, |\, w(a)>q \}$ , as before.

A strict partition $\lambda = (\lambda _1>\cdots >\lambda _s>0)$ determines a Grassmannian signed permutation $w=w_\lambda $ by setting $w(i)=1-\lambda _i$ for $1\leq i\leq s$ , and filling in the remaining unused values in increasing order. For example, $\lambda = (4,2,1)$ has Grassmannian signed permutation $w_\lambda =[-3,-1,0,3]$ . Schubert varieties $\Omega _\lambda \subseteq \mathrm {LG}$ are defined by conditions $\dim ( E \cap V_{>\lambda _k} )\geq k$ .

As before, Schubert varieties in $\mathrm {Fl}^C$ determine unique Schubert classes. The (twisted) double Schubert polynomial of type C is the polynomial such that

$$\begin{align*}\boldsymbol{S}_w^C(c;x;y) = [\Omega_w] \end{align*}$$

under $\mathbf {\Gamma }[x_+,y_+] = H_{\mathbf {T}}^*\mathrm {Fl}^C$ . For $z=y=0$ , this is precisely the definition in [Reference Billey and Haiman8]; for $z=0$ , these are the double Schubert polynomials of [Reference Ikeda, Naruse and Mihalcea16]. Among the many wonderful properties of these polynomials, we mention the Cauchy formula:

(12) $$ \begin{align} \boldsymbol{S}_w^C(\mathbf{c};x;y) &= \sum_{uv \dot{=} w} \boldsymbol{S}_v^C(c;x;t) \, \boldsymbol{S}_u^C(c';z-t;y), \end{align} $$

where $\mathbf {c}=c\cdot c'$ .

One can compare Schubert polynomials in types A and C via the canonical surjection $\Lambda [z][x,y] \to \mathbf {\Gamma }[x_+,y_+]$ : for $w\in \mathcal {S}_+\subset W_\infty $ , this map sends $\boldsymbol {S}_w^A(c;x;y)$ to $\boldsymbol {S}_w^C(c;x;y)$ . A geometric proof is in [Reference Anderson and Fulton5].

The twisted double Q -polynomials $\boldsymbol {Q}_\lambda (c|y) = \boldsymbol {S}_{w_\lambda }^C(c;x;y)$ correspond to Schubert classes in $\mathrm {LG}$ , so they form a basis for $\mathbf {\Gamma }[y_+]$ over $\mathbb {Z}[z][y_+]$ . At $z=0$ (and an appropriate evaluation of c), these specialize to Ivanov’s double Q-functions; at $z=y=0$ , they specialize to Schur’s Q-polynomials $Q_\lambda (c)$ , which form a basis for $\Gamma $ .

9.3 Direct sum and coproduct

The embedding $\mathrm {LG} \subset \mathrm {Gr}$ is compatible with the direct sum map, where one takes the symplectic form on $\mathbb {V}=V\oplus V$ to be the difference of symplectic forms on each summand. So one obtains a coproduct $\Delta \colon \mathbf {\Gamma }[y_+] \to \mathbf {\Gamma }[y_+]\otimes _{\mathbb {Z}[y]} \mathbf {\Gamma }[y_+]$ . Similarly, the direct sum morphism $\mathrm {LG}(V) \times \mathrm {Fl}^C(V) \to \mathrm {Fl}^C(\mathbb {V})$ determines a co-module homomorphism $\mathbf {\Gamma }[x_+;y_+] \to \mathbf {\Gamma }[y_+]\otimes _{\mathbb {Z}[y]} \mathbf {\Gamma }[x_+,y_+]$ .

In Schubert classes, we can again write

$$\begin{align*}\boxplus^*[\Omega_w] = \sum_{\mu,v} \widehat{f}_{\mu,v}^w(y;z) [\Omega_u] \times [\Omega_v], \end{align*}$$

for strict partitions $\mu $ and signed permutations $v,w$ , where the polynomials $\widehat {f}_{\mu ,v}^w(y;z)$ are type C double Edelman-Greene coefficients.

Using Cauchy formulas, this co-module operation on Schubert polynomials can be written as

$$ \begin{align*} \boldsymbol{S}_w^C(\mathbf{c};x;y) &= \sum_{uv\dot{=}w} \boldsymbol{F}^C_u(c;y) \cdot \boldsymbol{S}_v^C(c';x;y) \\ &= \sum_{\mu,v} \widehat{f}_{\mu,v}^w(y;z) \, \boldsymbol{Q}_\mu(c|{y}) \, \boldsymbol{S}_v^C(c';x;y) , \end{align*} $$

where the (twisted) double type C Stanley polynomial is defined as

$$\begin{align*}\boldsymbol{F}^C_w(c;y) = \boldsymbol{S}^C_w(c;z-y;y). \end{align*}$$

As before, the coefficients $\widehat {f}_{\mu ,v}^w(y;z)$ arise in the expansion of $\boldsymbol {F}^C_{wv^{-1}}$ in the $\boldsymbol {Q}_\mu $ basis.

Also as before, the direct sum morphism is actually equivariant with respect to the larger $T \times T\times (\mathbb {C}^*)$ action on $\mathbb {V}=V\oplus V$ , where the $\mathbb {C}^*$ factor still acts diagonally (though once again, the extended equivariant structure does not define a commutative coproduct). Writing $y_i$ for the characters on the first factor and $y^{\prime }_i$ for those on the second factor, we can expand

$$\begin{align*}\boxplus_*[X_\mu \times X_v] = \sum_{\mu,v}\widehat{f}_{\mu,v}^w(y,y';z) \, [\mathbf{X}_w] \end{align*}$$

in $H_{T \times T\times (\mathbb {C}^*)}^*\mathrm {Fl}^C(\mathbb {V})$ .

The proof is the same as for Theorems 8.5 and 8.7, applying Graham’s theorem and keeping track of weights on the corresponding unipotent groups in symplectic groups.

Specializing $y=y'$ , one obtains the type C analogue of a weak form of Theorem 8.5: $\widehat {f}_{\mu ,v}^w(y;z)$ is a nonnegative sum of squarefree monomials in $-y_i-y_j+z$ and $y_i-y_j$ . This version requires no appeal to a quantum-affine isomorphism, which was used in the above proof of Theorem 8.5 to show that only $y_i-y_j$ with $i>j$ can appear in $\widehat {c}_{\mu ,v}^w(y)$ . It should be interesting to adapt the methods of [Reference Anderson1] to establish a stronger positivity statement for $\widehat {f}_{\mu ,v}^w(y;z)$ , analogous to that of Theorem 8.5.

Remark 9.4. In the Lagrangian Grassmannian case where $w=w_\lambda $ and $v=w_\nu $ for strict partitions $\lambda $ and $\nu $ , the polynomial $\widehat {f}_{\mu ,\nu }^\lambda (y)$ may be regarded as a dual Hall-Littlewood coefficient. It expresses the coproduct

$$\begin{align*}\boldsymbol{Q}_\lambda(\mathbf{c}|y) = \sum_{\mu,\nu} \widehat{f}_{\mu,\nu}^\lambda(y;z)\, \boldsymbol{Q}_\mu(c|y) \cdot \boldsymbol{Q}_\nu(c'|y), \end{align*}$$

where $\mathbf {c}=c\cdot c'$ as usual. Evaluating at $y=z=0$ , this is the structure constant for multiplication in the basis of P-Schur functions; that is, $\widehat {f}_{\mu ,\nu }^\lambda (0) = f_{\mu ,\nu }^\lambda $ in the notation of [Reference Macdonald23, §III.5]. Combinatorial formulas for this case were given by Stembridge [Reference Stembridge26].