%Date: Sun, 6 Dec 1998 21:01:15 +0200
%From: Boris Shapiro <shapiro@matematik.su.se>

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\title{Algebras of Curvature Forms\\ on Homogeneous Manifolds}
\author{%
Alexander Postnikov\\[.05in]
{\ns M.I.T., Cambridge, MA 02139, U.S.A.}\\[.2in]
Boris Shapiro\\[.05in]
{\ns University of Stockholm, Stockholm, S-10691, Sweden} \\[.2in]
Mikhail Shapiro\\[.05in]
{\ns Royal Institute of Technology, Stockholm, S-10044, Swedeon}\\[.2in]}

\date{March 4, 1998}

\begin{document}

\maketitle


\begin{abstract}
Let~$\CC(X)$ be the algebra generated by the curvature two-forms of standard
holomorphic hermitian line bundles over the complex homogeneous manifold
$X=G/B$.
The cohomology ring of~$X$  is a quotient of~$\CC(X)$.  We calculate
the Hilbert polynomial of this algebra.  In particular, we show that the
dimension of~$\CC(X)$ is equal to the number of independent subsets of roots in
the corresponding root system.
\end{abstract}




\section{Homogeneous Manifolds}


In this section we remind the reader the basic notions and notation related to
homogeneous manifolds~$G/B$ and root systems, as well as fix our terminology.
\medskip


Let~$G$ be a connected complex semisimple Lie group and~$B$ its Borel
subgroup.  The quotient space~$X=G/B$ is then a compact homogeneous complex
manifold.  We choose a maximal compact subgroup~$K$ of~$G$ and denote by
$T=K\cap B$ its maximal torus.  The group~$K$ acts transitively on~$X$.
Thus~$X$ can be identified with the quotient space~$K/T$.
\medskip

By~$\g$ we denote the Lie algebra of~$G$ and by~$\h\subset\g$ its Cartan
subalgebra.   Also denote by~$\g_\R\subset\g$ the real form of~$\g$ such
that $i\,\g_\R$ is the Lie algebra of $K$.  Analogously,
$\h_\R=\h\cap\g_\R$ and
$i\,\h_\R$ is the Lie algebra of the maximal torus $T$.
%Then~$\g$ is the complexification
%of~$\g_R$ is the complexified Lie algebra of the maximal torus~$T$.
The {\it root system} associated with~$\g$ is the set~$\Delta$ of nonzero
vectors (roots)~$\alpha\in\h^*$ for which the root spaces
$$
\g_\alpha = \{x\in\g \mid [h,x]=\alpha(h) x\textrm{ for all }h\in\h\}
$$
are nontrivial.  Then $\g$ decomposes into the direct sum of subspaces
$$
\g = \h\oplus\sum_{\alpha\in\Delta} \g_\alpha.
$$
For $\alpha\in\Delta$, the spaces~$\g_\alpha$ and $\h_\alpha=
[\g_\alpha,\g_{-\alpha}]$ are one-dimensional and there exists a unique element
$h_\alpha\in\h_\alpha$ such that $\alpha(h_\alpha)=2$.  The elements
$h_\alpha\in\h$ are called {\it coroots.}  Actually, $\alpha\in\h_\R^*$ and
$h_\alpha\in
\h_\R$, for $\alpha\in\Delta$.  Let us choose generators
$e_\alpha\in\g_\R$ of the root spaces $\g_\alpha$ such that
$[e_\alpha,e_{-\alpha}]=h_\alpha$ for any root $\alpha$.
Then $[h_\alpha,e_\alpha]=2e_\alpha$ and
$[h_\alpha,e_{-\alpha}]=-2e_{-\alpha}$.


The root system~$\Delta$ is subdivided into a disjoint union of sets of
positive
roots~$\Delta_+$ and negative roots~$\Delta_-=-\Delta_+$ such that the direct
sum $\mathfrak{b}=\h\oplus\sum_{\alpha\in\Delta_+}\g_\alpha$ is the Lie algebra
of Borel subgroup~$B$.
The Weyl group~$W$ is the group generated by the
reflections~$s_\alpha:\h^*\to\h^*$,
$\alpha\in\Delta_+$, given by
$$
s_\alpha: \lambda \mapsto \lambda -  \lambda(h_\alpha)\, \alpha.
$$





The lattice $\hat{T}=\{\lambda\in\h^*\mid \lambda(h_\alpha)\in\Z
\textrm{ for all } \alpha\in\Delta\}$ is called the {\it weight lattice.}
Every weight $\lambda\in\hat{T}$ determines an irreducible unitary
representations~$\pi_\lambda: T\to\C^\times$ of the maximal torus~$T$ given by
$\pi_\lambda(\exp(x))=e^{\lambda(x)}$, for $x\in i\,\h_\R$, and every
irreducible
unitary representation of~$T$ is of this form.



For a weight~$\lambda\in\hat{T}$, the
homomorphism~$\pi_\lambda:T\to\C^\times$
extends uniquely to a holomorphic homomorphism~$\bar\pi_\lambda:
B\to\C^\times$.  Thus any~$\lambda\in\hat{T}$ determines a holomorphic line
bundle~$L_\lambda = G\times_B \C = K\times_T\C$ over $X=G/B=K/T$.
The line bundle~$L_\lambda$ has a canonical $K$-invariant hermitian metric.

The classical Borel's theorem~\cite{borel} describes the cohomology
ring~$\H^*(X,\C)$ of the homogeneous manifold~$X$ in terms of generators
and relations:
$$
\H^*(X,\C)\cong \Sym(\h^*)/ I_W,
$$
where~$I_W$ is the ideal in the symmetric algebra~$\Sym(\h^*)$ generated by
the~$W$-invariant elements without constant term.  The natural projection
from~$\Sym(\h^*)$ to~$\H^*(X,\C)$ is the homomorphism that sends a weight
$\lambda\in\hat{T}$ to the first Chern class~$c_1(L_\lambda)$ of the
line bundle~$L_\lambda$.

\medskip
The purpose of this article is to extend the cohomology ring~$\H^*(X,\C)$
to the level of differential forms on~$X$.
It is possible to exhibit differential two-forms that represent the Chern
classes~$c_1(L_\lambda)$ in the de~Rham cohomology of the homogeneous
manifold~$X$.
Recall that for a holomorphic hermitian line bundle~$L:E\to X$ there is
a canonically associated connection on~$E$.  Denote by $\Theta(L)$  the {\it
curvature form} of this connection, which is a differential two-form on~$X$.
Then the form~$i\,\Theta(L)/2\pi$ represents~$c_1(L)$.

In order to construct the curvature forms~$\Theta(L_\lambda)$ explicitly, we
define the elements~$e^\alpha\in\g_\R^*$, $\alpha\in\Delta$, by
$e^{\alpha}(\h)=0$ and $e^\alpha(e_\beta)=\delta_{\alpha\beta}$, for any
$\beta\in\Delta$.  (Here~$\delta_{\alpha\beta}$ is Kroneker's delta.)  The
space of left $K$-invariant differential one-forms on~$K$
can be identified with the dual to its
Lie algebra, t.e., with~$i\,\g_\R^*$.  Thus the elements $i\,e^\alpha$ can be
regarded as one-forms on~$K$.  The differential two-form on~$K$ given by
$$
\phi_\alpha=e^\alpha\wedge e^{-\alpha},\quad \alpha\in\Delta,
$$
is invariant with respect to the right translation action of the torus~$T$.
Thus~$\phi_\alpha$ produces a two-form on the manifold~$X$, for which we
will use the same notation~$\phi_\alpha$.
It is clear from the definition that $\phi_{-\alpha}=-\phi_\alpha$.

The following statement is implicit in~\cite{GS}.

\begin{proposition}  For $\lambda\in\hat{T}$, the curvature form
of the holomorphic hermitian line bundle~$L_\lambda$ is given by
$$
\Theta(L_\lambda) = \sum_{\alpha\in\Delta_+} \lambda(h_\alpha)\,\phi_\alpha.
$$
\end{proposition}

Let~$\Phi$ be the algebra generated by the two-forms~$\phi_\alpha$,
$\alpha\in\Delta_+$.
The relations in~$\Phi$ are easy:
$$
\phi_\alpha \phi_\beta = \phi_\beta \phi_\alpha,\\
\qquad\quad\phi_\alpha^2=0.
$$
Thus $\Phi$ is a $2^N$-dimensional algebra, where~$N=|\Delta_+|$.
The main object in this paper is the subalgebra of~$\Phi$ generated
by the curvature forms~$\Theta(L_\lambda)$.


\section{Main Results}

Denote by~$\CC(X)$ the subalgebra in the algebra of differential forms on~$X$
that is generated by the curvature forms~$\Theta(L_\lambda)$ of line bundles.
Obviously, $\CC(X)$ has the structure of a graded ring:
$\CC(X)=\CC^0(X)\oplus\CC^1(X)\oplus \CC^2(X)\oplus\cdots$,
where $\CC^k(X)$ is the subspace of~$2k$-forms
in~$\CC(X)$.  In order to formulate our main results about~$\CC(X)$ we need
some extra notation.

Let~$V$ be a collection of vectors~$v_1,v_2,\dots,v_N$ in
a vector space~$E$.  A subset of vectors in $V$ is called {\it independent} if
they are linearly independent in~$E$.  By convention an empty subset is
independent.
Let~$\ind(V)$ be number of all independent subsets in~$V$.

A {\it cycle\/} is a minimal by inclusion not independent subset.
For a cycle $C=\{v_{c_1},\dots,v_{c_l}\}$, there is a unique, up to a factor,
linear dependence $\lambda_1 v_{c_1}+\dots +\lambda_l v_{c_l}=0$
with non-zero $\lambda_i$'s.  Let us fix a linear order
$v_1<v_2<\cdots<v_N$ of all elements of~$V$.
For an independent subset $S$, a vector~$v\in V\setminus S$,
is called {\it externally active\/} if $S\cup \{v\}$ contains a cycle $C$
and $v$ is the minimal element of~$C$.  Let $\act(S)$ be the number
of externally active vectors.

%We assume now that $E=\h^*$ and $V=\Delta_+$ is the collection of positive
roots.
\begin{theorem}
\label{th:main}
The dimension of the algebra $\CC(X)$ is equal to the number
$\ind(\Delta_+)$ of
independent subsets in the set of positive roots~$\Delta_+$.  Moreover, the
dimension of the $k$-th component $\CC^k(X)$ is equal to the number of
independent subsets~$S\subset\Delta_+$ such that $k=N-|S|-\act(S)$,
where~$N=|\Delta_+|$.
\end{theorem}

We remark here that, although the number $\act(S)$ of externally active vectors
depends upon a particular order of elements in~$V$, the total number of
subsets~$S$ with fixed~$|S|+\act(S)$ does not depend upon a choice of ordering.


We will actually prove a more general result about an arbitrary collection of
vectors~$V$.  Let us choose an inner product $\<\cdot,\cdot\>$ and an
orthogonal basis~$\ep_1,\dots,\ep_n$ in~$E$.
Denote by $\Phi_N$ the algebra generated by $N$ commuting
generators~$\phi_1,\phi_2,\dots,\phi_N$ with relations $\phi_i^2=0$.
Let~$\cC(V)$ be the subalgebra in~$\Phi_N$
generated by the elements
$$
\theta_i=\sum_{j=1}^N \<\ep_i,v_j\>\, \phi_j,\qquad
i=1,\dots,n.
$$
The algebra $\cC(V)$ has an obvious grading~$\cC(V)=\cC^0(V)\oplus
\cC^1(V)\oplus \cC^2(V)\oplus \cdots$ by degree of elements.

If~$E=\h$ and $V$ is the collections of coroots~$h_\alpha$,
$\alpha\in\Delta_+$,
then~$\cC(V)=\CC(X)$.  In general, we have the following result.


\begin{theorem}
\label{th:AV}
The dimension of the algebra $\cC(V)$ is equal to the number
$\ind(V)$ of independent subsets in~$V$.  Moreover, the dimension of
the $k$-th component $\cC^k(V)$ is equal to the number of
independent subsets~$S\subset V$ such that $k=N-|S|-\act(S)$.
\end{theorem}


\section{Dual Statement and Proof}

Next we give a dual version of Theorem~\ref{th:AV}.  Let~$\Sym(E)$ be the
symmetric algebra of the vector space~$E$.  For a subset $S\subset V$, denote
$\<S\> =\prod_{v\in S} v\in\Sym(E)$, a square-free monomial in~$v_i$'s.  Let
us denote by $\S(V)$ a subspace in $\Sym(E)$ spanned by $2^N$ square-free
monomials
$\<S\>$, where $S$ ranges over all subsets in $V$.  Let $\S^k(V)$ be the $k$-th
graded component of~$\S(V)$.

\begin{theorem}
\label{th:independent}
The dimension of the subspace $\S(V)$ in $\Sym(E)$ is equal to the number
$\ind(V)$ of independent subsets in~$V$.
Moreover, the dimension of~$\S^k(V)$ is
equal to the number of independent subsets $S\subset V$ such that $k =
N-|S|-\act(S)$.
\end{theorem}

For $I=(1\leq i_1<i_2<\dots<i_k\leq N)$ and $J=(1\leq j_1\leq j_2 \leq  \cdots
\leq j_k\leq n)$, we denote by $a_{IJ}$ the coefficient of the square-free
monomial $\phi_{i_1}\cdots \phi_{i_k}\in \Phi_N$ in the product of generators
$\theta_{j_1}\cdots \theta_{j_k}\in \cC(V)$.  Let $A_k=(a_{IJ})$ be the
${N\choose k}
\times {n+k-1\choose k}$-matrix formed by the~$a_{IJ}$.  Clearly, $\dim
\cC^k(V)$ is equal to the rank of the matrix $A_k$.

On the other hand, $a_{IJ}$ is also the coefficient of the monomial
$\ep_{j_1}\cdots
\ep_{j_k}\in \Sym(E)$ in the square-free monomial $v_{i_1}\cdots v_{i_k}\in
\S(V)$.  Thus $\dim \S^k(V)$ is equal to the rank of the transposed matrix
$A_k^T$, which is the same as the rank of~$A_k$.  This shows that
$\dim\cC^k(V)=\dim\S^k(V)$ and
Theorem~\ref{th:AV} is equivalent to Theorem~\ref{th:independent}.

\medskip


We say that a subset $S\subset V$
is {\it robust\/} if there is no cycle~$C\subset V$ with minimal element~$v$
such that $S\cap C=\{v\}$.
\begin{lemma}
\label{le:NBC}
The number of robust subsets in~$V$ is equal to the number of independent
subsets.
Moreover, the number of $k$-element robust subsets is equal to
the number of independent subsets $S$ with $k=N-|S|-\act(S)$.
\end{lemma}

\proof
We present an explicit bijection between robust subsets and independent
subsets.  A subset~$A$ is the complement to a robust subset in~$V$
if and only if for any cycle $C$ with minimal element $v$, inclusion
$C\setminus\{v\}\subset A$ implies $C\subset A$.  We will call such subsets
{\it antirobust.}

For a independent subset $S$, let $M$ be the collection of all
externally active $v\in V\setminus S$.  Then~$S\cup M$ is antirobust.
%satisfies the above property. Thus its complement $V\setminus
%(S\cup K)$ is a robust subset.
Conversely, for an antirobust subset $A$, let $M$ be the collection of all
$v\in V$ such that $v$ is a minimal element in some cycle $C\subset A$.  Then
$A\setminus M$ is an independent subset.

Clearly, both these mapping are inverse to each other and the statement of
lemma follows.
\endproof


\begin{theorem}
\label{th:robust}
The set of square-free monomials~$\<S\>$, where $S$ ranges over
robust subsets, forms a basis of the subspace $\S(V)$.
\end{theorem}

First, we prove a weaker version of Theorem~\ref{th:robust}.

\begin{lemma}
\label{le:robust}
The square-free monomials~$\<S\>$, where $S$ ranges over
robust subsets, span $\S(V)$.
\end{lemma}

\proof Suppose not.  Let $\<R\>$ be the maximal in the lexicographical order
square-free monomial which cannot be expressed linearly via the
monomials~$\<S\>$ with robust~$S$.  Then there is a cycle
$C=\{v_{c_1},v_{c_2},\dots,v_{c_l}\}$ with the minimal element~$v=v_{c_1}$ such
that $R\cap C=\{v\}$.  We can replace $v$ in the monomial $\<R\>$ by a linear
combination of $v_{c_2}, v_{c_3},\dots,v_{c_l}$.  Thus $\<R\>$ is a linear
combination of square-free monomials which are greater than $\<R\>$ in the
lexicographical order.  By assumption each of these monomials can be expressed
via the monomials $\<S\>$ with robust $S$.  Contradiction.  \endproof


We can now conclude the proof.

\proofof{Theorems~\ref{th:independent} and~\ref{th:robust}}
Lemmas~\ref{le:NBC} and~\ref{le:robust} imply the inequality $\dim
\S(V)\leq \ind(V)$.
In view of these two lemmas it is enough to show that $\dim \S(V)$ is
actually equal
to $\ind(V)$.

We prove this statement by induction on $|V|$.  If the linear span of
vectors in~$V$
is one-dimensional, then both $\dim \S(V)$ and $\inv(V)$ are equal to
the number of non-zero vectors in~$V$ plus one.  This establishes the base
of induction.

Assume that $v=v_N$ is a nonzero vector in~$V$.
Let $V'=V\setminus\{v\}=\{v_1,\dots,v_{N-1}\}$, and
let $V''$ be the collection of images of $v_1, \dots, v_{N-1}$
in the quotient space $E/\<v\>$.  It is immediate from the definition of
independent subset that $\ind(V) = \ind(V') + \ind(V'')$.
Assume by induction that $\dim \S(V')=\ind(V')$ and
$\dim \S(V'')=\ind(V'')$.

Clearly, $\S(V)$ is spanned by $\S(V')+\S(V')v$.   Both $\S(V')$ and $\S(V')v$
have same dimensions.  Hence, $\dim \S(V)= 2 \dim \S(V')-\dim (\S(V')\cap
\S(V')v)$.  Let $p:\S(V')\to \S(V'')$ be the natural projection.  Then
$\S(V')\cap \S(V')v\subset \Ker(p)$.  Thus $\dim \S(V') - \dim (\S(V')\cap
\S(V')v) \geq  \dim \S(V'')$ and $\dim \S(V)\geq \dim \S(V')+\dim \S(V'')
=\ind(V')+\ind(V'')=\ind(V)$.  Coupled with the inequality $\dim \S(V)\leq
\ind(V)$, this produces the required statement.

This finishes the proof of Theorems~\ref{th:independent} and~\ref{th:robust}
and thus of Theorems~\ref{th:main} and~\ref{th:AV}.
\endproof






\section{Remarks and Open Problems}

The algebra $\CC(X)$ for type~A flag manifolds $X=SL(n)/B$ was studied in
more details in~\cite{SS} and~\cite{PSS}.  In this case, we first proved
Theorem~\ref{th:main} in~\cite{PSS} using a different approach based on a
presentation of~$\CC(X)$ as a quotient of a polynomial ring (cf.~conjecture
below).  The theorem claims that the dimension of~$\CC(X)$ is equal to the
number forests on~$n$ labelled vertices whereas the dimension of~$\CC^k(X)$ is
equal to the number of forests with ${n \choose 2} - k$ inversions.  The
statement about forests was initially conjectured in~\cite{SS} and the
statement concerning inversions was then guessed by R.~Stanley.  In~\cite{PSS}
we also discuss various generalizations of the ring $\CC(X)$.

\medskip

There is a natural projection $\Sym(\h^*)\to \CC(X)$ that send
$\lambda\in\hat{T}$
to the curvature form $\Theta(L_\lambda)$.  Let~$J$ be the kernel of this
homomorphism.  Then $\CC(X)$ is naturally isomorphic to $\Sym(\h^*)/J$.  It
would
be interesting to describe the ideal~$J$ explicitly.
Let $\omega_1,\omega_2,\dots,\omega_l$ be the fundamental weights.  They
generate the weight lattice~$\hat{T}$.  Also let $d_i$ be the number of
positive
roots $\alpha\in\Delta_+$ such that $\alpha(\omega_i)\ne 0$.

\begin{conjecture}  The ideal~$J\in\Sym(\h^*)$ is generated by the elements
$(w\cdot \omega_i)^{d_i+1}$, where $i=1,\dots,l$ and $w$ is an element of the
Weyl group~$W$.
\end{conjecture}
It is clear from a dimension argument that~$(w\cdot \omega_i)^{d_i+1}\in J$.
The nontrivial part is to show that these elements generate the ideal~$J$.
This statement was proved for type~$A$ flag manifolds in~\cite{SS}.

\medskip

Another natural open problem is to extend the results to homogeneous manifolds
$G/P$, where~$P$ is a parabolic subgroup.
Formulas for curvature forms on~$G/P$ can be found in \cite{GS}.

\medskip

It is also intriguing to find the links between the algebra~$\CC(X)$ and
the arithmetic Schubert calculus developed in~\cite{Ta1, Ta2}.


\medskip

B.~Kostant pointed out that the algebra~$\CC(X)$ is related to the
$S(\g)$-module $\wedge\g$ studied in~\cite{K}.  It would be interesting to
investigate this relationship.




\medskip



\bigskip
The authors are  grateful to Richard Stanley, Alek Vainshtein,
and Andrei Zelevinsky for stimulating discussions.


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\bibitem{GS} P.~Griffiths and W.~Schmid, Locally homogeneous complex manifolds,
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\bibitem{K} B.~Kostant, On $\wedge\g$ for a semisimple Lie algebra~$\g$,
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\bibitem{PSS} A.~Postnikov, B.~Shapiro, and M.~Shapiro, Chern forms on flag
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\bibitem{Ta2} H.~Tamvakis, Arithmetic intersection theory on flag varieties,
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\end{thebibliography}

\end{document}
