Fukukawa, Yukiko

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TitleTHE COHOMOLOGY RING OF THE GKM GRAPH OF AFLAG MANIFOLD (Transformation Groups and SurgeryTheory)Author(s)Fukukawa, YukikoCitation数理解析研究所講究録 (2011), 1732: 86-91Issue Date2011-03URL http://hdl.handle.net/2433/170593RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityTHE COHOMOLOGY RING OF THE GKM GRAPHOF A FLAG MANIFOLD(Yukiko Fukukawa)Department ofMathematics,Osaka City University1. INTRODUCTIONLet $T$ be a toms of dimension $n$ and $M$ a closed smooth T-manifold.The equivariant cohomology of $M$, denoted $H_{T}^{*}(M)$ , contains a lot of ge-ometrial information on $M$. Moreover it is often easier to compute $H_{T}^{*}(M)$than $H^{*}(M)$ by virtue of the Localization Theorem which implies that therestriction map(1.1) $\iota^{*}:H_{T}^{*}(M)arrow H_{T}^{*}(M^{T})$to the T-fixed point set $M^{T}$ is often injective, in fact, this is the case when$H^{odd}(M)=0$ . When $M^{T}$ is isolated, $H_{T}^{*}(M^{T})=\oplus_{p\in M^{T}}H_{T}^{*}(p)$ and hence$H_{T}^{*}(M^{T})$ is a direct sum ofcopies ofa polynomial ring in $n$ variables because$H_{T}^{*}(p)=H^{*}(BT)$ .Therefore we are in a nice situation when $H^{odd}(M)=0$ and $M^{\Gamma}$ is iso-lated. Goresky-Kottwitz-MacPherson [2] (see also [3, Chapter 11]) foundthat under the hrther condition that the weights at a tangential T-moduleare pairwise linearly independent at each $p\in M^{T}$ , the image of $\iota$ “ in (1.1)above is determined by the fixed point sets ofcodimension one subtori of $T$when $\mathbb{Q}$ is tensored in cohomology. Their result motivated Guillemin-Zara[4] to associate a labeled graph $\mathcal{G}_{M}$ with $M$ and define the “cohomology”ring $H^{*}(\mathcal{G}_{M})$ of $\mathcal{G}_{M}$ , which is a subring of $\oplus_{p\in M^{T}}H^{*}(BT)$ . Then the result ofGoresky-Kottwitz-MacPherson can be stated that $H_{T}^{*}(M)\otimes \mathbb{Q}$ is isomorphicto $\mathcal{H}^{*}(\mathcal{G}_{M})\otimes \mathbb{Q}$ as graded rings when $M$ satisfies the conditions mentionedabove.The result of Goresky-Kottwitz-MacPherson can be applied to many im-portant T-manifolds $M$ such as flag manifolds and compact smooth toricvarieties etc. When $M$ is such a nice manifold, $H_{T}^{*}(M)$ is often known tobe isomolphic to $\prime H^{*}(\mathcal{G}_{M})$ without tensoring with $\mathbb{Q}$ (see [1], [5], [6] forexample). We determine the ring structure of $H^{*}(\mathcal{G}_{M})$ or $H^{*}(G_{M}) \otimes Z[\frac{1}{2}]$when $M$ is a flag manifold of classical type directly without using the fact1732 2011 86-91 86that $H_{T}^{*}(M)$ is isomorphic to $H^{*}(\mathcal{G}_{M})$ ([7]). In my talk, I introduced the re-sult when $M$ is a flag manifold oftype A. This is ajoint work with HiroakiIshida and Mikiya Masuda and the details can be found in [7].2. LABELED GRAPH AND ITS COHOMOLOGY FOR TYPE $A_{n-1}$Let $\{t_{i}\}_{i=1}^{n}$ be a basis of $H^{2}(BT)$ , so that $H^{*}(BT)$ can be identified with apolynomial ring $Z[t_{1}, t_{2}, \ldots, t_{n}]$ . We take an inner product on $H^{2}(BT)$ suchthat the basis $\{t_{i}\}$ is orthonormal. Then(2.1) $\Phi(A_{n-1})$ $:=\{\pm(t_{i}-t_{j})|1\leq i<j\leq n\}$is a root system of type $A_{n-1}$ .Definition. The labeled graph associated with $\Phi(A_{n-1})$ , denoted $\ovalbox{\tt\small REJECT}_{n}$ , is agraph with labeling $f$ defined as follows.$\bullet$ The veltex set of $\ovalbox{\tt\small REJECT}_{n}$ is the permutation group $S_{n}$ on $\{1,2,\ldots,n\}$ .$\bullet$ Two vertices $w,$ $w’$ in $\ovalbox{\tt\small REJECT}_{n}$ are connected by an edge $e_{w,w’}$ if and onlyif there is a transposition $(i,j)\in S_{n}$ such that $w’=w(i,j)$, in otherwords,$w’(i)=wC),$ $w’(J)=w(i)$ and $w’(r)=w(r)$ for $r\neq i,$ $j$.$\bullet$ The edge $e_{w,w’}$ is labeled by $f(e_{w,w’}):=t_{w(i)}-t_{w’(i)}$ .Definition. The cohomology ring of $ffl_{n}$ , denoted $\mathcal{H}^{*}(it_{n})$, is defined tobe the subring of Map$(V(J71_{n}),H^{*}(BT))=\oplus_{v\in V(\mathcal{J}1_{n})}H^{*}(BT)$ , where $V(PI_{n})$denotes the set of veltices of $9t_{n}$ , i.e. $V(\ovalbox{\tt\small REJECT} I_{n})=S_{n}$ , satisfying the followingcondition:$f\in$ Map$(V(\ovalbox{\tt\small REJECT}_{n}), H^{*}(BT))$ is an element of $H^{*}(\ovalbox{\tt\small REJECT} l_{n})$ if andonly if $f(v)-f(v’)$ is divisible by $f(e)$ in $H^{*}(BT)$ wheneverthe veltices $v$ and $v’$ are connected by an edge $e$ in $\ovalbox{\tt\small REJECT}_{n}$ .For each $i=1,$ $\ldots,$ $n$ , we define elements $\tau_{i},$ $t_{i}$ of Map$(V(Pl_{n}),H^{*}(BT))$by(2.2) $\tau_{i}(w)$ $:=t_{w(l)}$ , $t_{i}(w)$ $:=t_{i}$ for $w\in S_{n}$ .In fact, both $\tau_{i}$ and $t_{i}$ are elements of $H^{2}(\ovalbox{\tt\small REJECT}_{n})$ .Example. The case $n=3$ . The root system $\Phi(A_{2})$ is $\{\pm(t_{i}-t_{j})|1\leq i<j\leq$$3\}$ . The labeled graph $\ovalbox{\tt\small REJECT}_{3}$ and $\tau_{i}$ for $i=1,2,3$ are as follows.123 12-13132231 321 $t_{2}$ $t_{3}$ $t_{3}$ $t_{2}$ $t_{1}$ $t_{1}$The labclcd graph $\ovalbox{\tt\small REJECT}_{3}$ $\tau_{1}$ $\tau_{2}$ $\tau_{3}$87Theorem 2.1. Let $\ovalbox{\tt\small REJECT}_{n}$ be the labeled graph associated with the root system$\Phi(A_{n-1})$ oftype $A_{n-1}$ in (2.1). Then$q-!^{*}(\ovalbox{\tt\small REJECT}_{n})=\mathbb{Z}[\tau_{1}, \cdots,\tau_{n}, t_{1}, \cdots, t_{n}]/(e_{i}(\tau)-e_{i}(t)|i=1, \cdots, n)$ ,where $e_{i}(\tau)$ (resp. $e_{i}(t)$) is the $i^{1h}$ elementary symmetricpolynomial in $\tau_{1},$ $\cdots,\tau_{n}$(resp. $t_{1},$ $\ldots,$ $t_{n}$).To prove this theorem, we need the following two lemmas.Lemma 2.2. $\mathcal{H}^{*}(\ovalbox{\tt\small REJECT}_{n})$ is generated by $\tau_{1},$ $\cdots,\tau_{n},$ $t_{1},$ $\cdots,$ $t_{n}$ as a ring.Proof. We shall prove the lemma by induction on $n$ . When $n=1,$ $H^{*}(\ovalbox{\tt\small REJECT}_{1})$is generated by $t_{1}$ since $\ovalbox{\tt\small REJECT}_{1}$ is a point; so the lemma holds.Suppose that the lemma holds for $n-1$ . Then it suffices to show that anyhomogenous element $f$ of $q\prec^{*}(\ovalbox{\tt\small REJECT}_{n})$ , say ofdegree 2$k$, can be expressed as apolynomial in $\tau_{i}$ ’s and $t_{i}’ s$ . For each $i=1,$ $\ldots,$ $n$ , we set$V_{i}:=\{w\in S_{n}|w(i)=n\}$and consider the labeled full subgraph Li of $ffl_{n}$ with $V_{i}$ as the vertex set.Note that Li can naturally be identified with $\ovalbox{\tt\small REJECT}_{n-1}$ for any $i$ .Let(2.3) $1 \leq q\leq\min\{k+1,n\}$and assume that(2.4) $f(v)=0$ for any $v\in V_{i}$ whenever $i<q$ .A vertex $w$ in $V_{q}$ is connected by an edge in $\ovalbox{\tt\small REJECT}_{n}$ to a vertex $v$ in $V_{i}$ ifand onlyif $v=w(i, q)$ . In this case $f(w)-f(v)$ is divisible by $t_{w(l)}-t_{w(q)}=t_{w(i)}-t_{n}$and $f(v)=0$ whenever $i<q$ by (2.4), so $f(w)$ is divisible by $t_{w(i)}-t_{n}$ for$i<q$ . Thus, for each $w\in V_{q}$ , there is an element $g^{q}(w)\in \mathbb{Z}[t_{1}, \cdots, t_{n}]$ suchthat(2.5) $f(w)=(t_{w(1)}-t_{n})(t_{w(2)}-t_{n})\ldots(t_{w(q-1)}-t_{n})g^{q}(w)$where $\mathscr{S}(w)$ is homogeneous and of degree $2(k+1-q)$ because $f(w)$ ishomogenous and of degree $2k$.One expresses(2.6) $g^{q}(w)= \sum_{r--0}^{k+1-q}g_{r}^{q}(w)f_{n}$with homogenous polynomials $g_{r}^{q}(w)$ ofdegree $2(k+1-q-r)$ in $Z[t_{1}, \cdots, t_{n-1}]$ .Then there is a polynomial $G_{r}^{q}$ in $\tau_{i}$ ’s (except $\tau_{q}$) and $t_{i}$ ’s (except $t_{n}$ ) such that$G_{r}^{q}(w)=g_{r}^{q}(w)$ for any $w\in V_{q}$ , because $g_{r}^{q}$ restricted $to.\mathcal{L}_{q}$ is an element of$H^{*}(.\mathcal{L}_{q})=7-!^{r}(\ovalbox{\tt\small REJECT}_{n-1}).\cdot$88Since $\tau_{i}(w)=t_{w(l)}$ and $w(i)=n$ for $w\in V_{i}$ , we have(2.7) $\prod_{i=1}^{q-1}(\tau_{i}-t_{n})(w)=0$ for any $w\in V_{i}$ whenever $i<q$ .Therefore, it follows from (2.5), (2.6), the Claim above and (2.7) that putting$G^{q}= \sum_{r--0}^{k+1-q}G_{r}^{q}t_{n}^{r}$ , we have$(f-G^{q} \prod_{i=1}^{q-1}(\tau_{i}-t_{n}))(w)=f(w)-g^{q}(w)\prod_{i=1}^{q-1}(t_{w(\iota)}-t_{n})$$=0$ for any $w\in V_{i}$ whenever $i\leq q$ .Therefore, subtracting the polynomial $G^{q} \prod_{i=1}^{q-1}(\tau_{i}-t_{n})$ from $f$, we mayassume that$f(v)=0$ for any $v\in V_{i}$ whenever $i<q+1$ .The above argument implies that $f$ finally takes zero on all vertices of $\ovalbox{\tt\small REJECT}_{n}$(which means $f=0$) by subtracting a polynomial in $\tau_{i}$ ’s and $t_{i}’ s$ , and thiscompletes the induction step. $\square$We abbreviate the polynomial ring $\mathbb{Z}[\tau_{1}, \cdots,\tau_{n}, t_{1}, \cdots, t_{n}]$ as $\mathbb{Z}[\tau, t]$ . Thecanonical map $Z[\tau, t]arrow\prime H^{*}(\ovalbox{\tt\small REJECT}_{n})$ is a grade preserving homomorphismwhich is $su\dot{q}$ ective by Lemma 2.2. Let $e_{j}(\tau)$ (resp. $e_{i}(t)$) denote the $i^{th}$elementary symmetric polynomial in $\tau_{1},$ $\cdots,\tau_{n}$ (resp. $t_{1},$ $\cdots,$ $t_{n}$). It easily fol-lows from (2.2) that $e_{i}(\tau)=e_{i}(t)$ for $i=1,$ $\cdots,$ $n$ . Therefore the canonicalmap above induces a grade preserving epimorphism(2.8) $Z[\tau, t]/(e_{1}(\tau)-e_{1}(t), \cdot\cdot\cdot, e_{n}(\tau)-e_{n}(t))arrow H^{*}(i1_{n})$.Remember that the Hilbert series of a graded ring $A^{*}=\oplus_{j=0}^{\infty}A^{j}$ , where $A^{j}$is the degree $j$ pall $ofA^{*}$ and offinite rank over $Z$, is a formal power seriesdefined by$F(A^{*}, s):= \sum_{j=0}^{\infty}(rank_{Z}A^{j})s^{j}$ .In order to prove that the epimorphism in (2.8) is an isomorphism, it sufficesto verify the following lemma because the modules in (2.8) are both torsionfree.Lemma 2.3. The Hilbert series of the both sides at (2.8) coincide, in fact,they are given by $\frac{1}{(1-s^{2})^{2n}}\prod_{i=1}^{n}(1-s^{2i})$ .Proof. (1) Calculation of LHS at (2.8). Let $e_{j}$ $:=e_{j}(\tau)-e_{i}(t)$ . It followsfrom the exact sequence$0arrow$ $(e_{1} , \cdot\cdot\cdot, e_{n})arrow Z[\tau, t]arrow Z[\tau, t]/(e_{1}, \cdots, e_{l})arrow 0$89that we have(2.9) $F(\mathbb{Z}[\tau, t]/(e_{1}, \cdots, e_{n}), s)=F(\mathbb{Z}[\tau, t],s)-F((e_{1}, \cdots, e_{n}), s)$.Here, since $\deg\tau_{i}=\deg t_{l}=2$ , we have(2.10) $F( \mathbb{Z}[\tau, t], s)=\frac{1}{(1-s^{2})^{2n}}$as easily checked; so it suffices to calculate $F((e_{1}, \cdots, e_{n}), s)$ .For $I\subset[n]$ we set $e_{I}$ $:= \prod_{i\in I}e_{i}$ . Then it follows from the Inclusion-Exclusion principle that(2.11) $F((e_{1}, \cdots, e_{n}), s)=\sum_{\emptyset\neq l\subset[n]}(-1)^{|I|-1}F((e_{I}), s)$and since $F((e_{I}), s)=s^{dcge_{J}}/(1-s^{2})^{2n}$ and $\deg e_{I}=\sum_{i\in I}2i$, it follows from(2.11) that(2.12) $F((e_{1}, \cdots, e_{n}), s)=\sum_{\phi\neq I\subset[n]}(-1)^{|J|-1}\frac{s^{\Sigma_{j\epsilon J}2i}}{(1-s^{2})^{2n}}$.Therefore it follows from(2.9), (2.10) and (2.12) that$F( \mathbb{Z}[\tau, t]/(e_{1}, \cdots, e_{n}), s)=\frac{1}{(1-s^{2})^{2n}}-\sum_{\psi\neq l\subset[n]}(-1)^{|l|-1}\frac{s^{\Sigma_{j\epsilon l}2i}}{(1-s^{2})^{2n}}$(2.13) $= \frac{1}{(1-s^{2})^{2n}}\sum_{I\subset[n]}(-1)^{|I|}s^{\Sigma_{j\epsilon l}2i}$$= \frac{1}{(1-s^{2})^{2n}}\prod_{i=1}^{n}(1-s^{2i})$ .(2) Calculation ofRHS at (2.8). Let $d_{n}(k)$ $:=rank_{Z}H^{2k}(\ovalbox{\tt\small REJECT}_{n})$ . Then(2.14) $F( \mathcal{H}^{*}(iI_{n}), s)=\sum_{k=0}^{\infty}d_{n}(k)s^{2k}$ .Recall the argument in the proof ofLemma 2.2. Since $g_{r}^{q}$ in (2.6) belongsto $H^{2(k+1-q-r)}(.\mathcal{L}_{q})=H^{2(k+1-q-r)}(\ovalbox{\tt\small REJECT}_{n-1})$ as shown in the Claim there, the rankof the module consisting ofthose $g^{q}$ in (2.5) and (2.6) is given by$\sum_{r=0}^{k+1-q}d_{n-1}(k+1-q-r)=\sum_{r=0}^{k+1-q}d_{n-1}(r)$ .Therefore, notin$g(2.3)$ , we see that the argument in the proof ofLemma 2.2implies$d_{n}(k)= \sum_{q--\iota}^{\min\{k+1,n|k}\sum_{r--0}^{+1-q}d_{n-1}(r)$ ,90in other words, ifwe set $d_{n-1}C$) $=0$ for $j<0$ , then(2.15)$d_{n}(k)=\{\begin{array}{ll}\sum_{i=1}^{n}id_{n-1}(k+1-i) if k\leq n-1,\sum_{i--\mathfrak{l}}^{n}id_{n-1}(k+1-i)+n\sum_{i=n+l}^{k+1}d_{n-1}(k+1-\iota) ifk \geq n.\end{array}$We shall abbreviate $F(H^{*}(\ovalbox{\tt\small REJECT}_{n}), s)$ as $F_{n}(s)$ . Then, plugging (2.15) in (2.14),we obtain$F_{n}(s)= \sum_{k=0}^{\infty}(d_{n-1}(k)+2d_{n-1}(k-1)+...$ $+nd_{n-1}(k+1-n))s^{2}$$+n \sum_{k=n}^{\infty}(d_{n-1}(k-n)+\cdots+d_{n-1}(1)+d_{n-1}(0))s^{2}$$=F_{n-1}(s)+2s^{2}F_{n-1}(s)+\cdot\cdot\cdot$ $+ns^{2n-2}F_{n-1}(s)$$+n(d_{n-1}(0)s^{2n} \frac{1}{1-s^{2}}+d_{n-1}(1)s^{2n+2}\frac{1}{1-s^{2}}+...)$$=F_{n-1}(s)(1+2s^{2}+ \cdots+ns^{2n-2})+n\frac{s^{2n}}{1-s^{2}}F_{n-1}(s)$$= \frac{1-s^{2n}}{1-s^{2}}F_{n-1}(s)$ .On the other hand, $F_{1}(s)=1/(1-s^{2})$ since $H^{*}(iI_{1})=Z[t_{1}]$ . It follows that$F_{n}(s)= \frac{1}{(1-s^{2})^{2n}}\prod_{i=1}^{n}(1-s^{2i})$ .This together with (2.13) proves the lemma. $\square$REFERENCES[1] A. Arabia, Cohomologie T-equivariante de $G/B$ pour an $c,roupeG$ de Kac-Moody, C.R. Acad. Sci. Paris S\’er. I. Math. 302 (1986) 631-634.[2] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul dualityand the localisation theorem, Invent. Math. 131 (1998) 25-83.[3] V. Guillemin and S. Stemberg, Supersymmetry and Equivariant de Rham Theory,Springer, $Berlin-Heidelberg$-New York, 1999.[4] V. Guillemin and C. Zara, l-skeleto, Betti numbers and equivariant cohomology, DukeMath. J. 107 (2001) 283-349.[5] M. Harada, A. Henriques and T. Holm, Computation ofgeneralized equivariant coho-mologies ofKac-Moodyflog varieties, Adv. in Math. 197 (2005), 198-221.[6] H. Maeda, M. Masuda and T. Panov, Torus graphs and simpliclal posets, Adv. Math.212 (2007), 458-483.[7] Y. Fukukawa, H. Ishida and M. Masuda, The cohomology ring ofthe $GKM$graph $ofa$flag manifold ofclassical type, preprint.91

THE COHOMOLOGY RING OF THE GKM GRAPH OF A FLAG MANIFOLD (Transformation Groups and Surgery Theory)

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THE COHOMOLOGY RING OF THE GKM GRAPH OF A FLAG MANIFOLD (Transformation Groups and Surgery Theory)

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