loop grassmaniann

dongmouren dongmouren Zhijie Dong

(February 2015; February 2015)

loop grassmaniann

dongmouren dongmouren Zhijie Dong

(February 2015; February 2015)

A relation between Mirković-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE

dongmouren dongmouren Zhijie Dong

(February 2015; February 2015)

Abstract

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the reduceness conjecture in [KMW16].

1 Introduction

Let $\mathfrak{g}$ be simply-laced semisimple finite dimensional complex Lie algebra. There are several modern constructions of irreducible representation of $\mathfrak{g}$ . In this paper we consider two models which realize the crystal of the positive part $U(\mathfrak{n})$ of $U(\mathfrak{g})$ . One is by Mirkovic-Vilonen (MV) cycles and the other is by the irreducible compotents of Lusztig’s nilpotent variety $\Lambda$ of the preprojective algebra of the quiver $Q$ corresponding to $\mathfrak{g}$ .

Baumann and Kamnitzer [BK12] studied the relations between $\Lambda$ and MV polytopes. They associate an MV polytope $P(M)$ to a generic module $M$ and construct a bijection between the set of irreducible components of $\Lambda$ and MV polytopes compatible with respect to crystal structures. Since MV polytopes are in bijection with MV cycles, Kamnitzer and Knutson launched a program towards geometric construction of the MV cycle $X(M)$ in terms of a module $M$ over the preprojective algebra.

Here we consider a version by Kamnizter, Knutson and Mirkovic: conjecturally, the ring of functions $\mathcal{O}(X(M)^{T})$ on the $T$ fixed point subscheme of the cycle $X(M)$ associted to $M$ , is isomorphic to $H^{*}(Gr^{\Pi}(M))$ , the cohomology ring of the quiver Qrassmannian of $M$ . In this paper I will construct a map from $\mathcal{O}(X(M)^{T})$ to $H^{*}(Gr^{\Pi}(M))$ and prove it is isomorphism for the case when $M$ is a representation of $Q$ .

In section two I recall the definition of MV cycles, quivers, preprojective algebra and Lusztig’s nilpotent variety and state the conjecture precisely.

In chapter three I describe the ring of functions on the $T$ fixed point subscheme of the intersection of closures of certain semi-infinite orbits (which is called ”cycle” in this paper). A particular case of these intersections is a scheme theoretic version of MV cycles. We realize these cycles as the loop Grassmannian with a certain condition Y.

In section four I construct the map $\Psi$ from $\mathcal{O}(X(M)^{T})$ to $H^{*}(Gr^{\Pi}(M))$ . Here, $\Psi$ maps certain generators of $\mathcal{O}(X(M)^{T})$ to Chern classes of tautological bundles over $Gr^{\Pi}(M)$ . So we need to check that the Chern classes satisfy the relations of generators of $\mathcal{O}(X(M)^{T})$ . We reduce this problem to a simple $SL_{3}$ case. In this case we have a torus action on $Gr^{\Pi}(M)$ so we could use localization in equivariant cohomology theory (GKM theory).

In chapter five I will prove $\Psi$ is an isomorphism in the case when $M$ is a representation of $Q$ of type A.

In chapter six I will state some consequences given the conjecture (one of which is the reduceness conjecture).

This is a piece of a big project to relate $\mathcal{G}(G)$ and the quiver $Q$ , see [Mir14], and more recently [Mir17b].

2 Statement of the conjecture

2.1 Notation

Let $G$ be a simply-laced semisimple group over complex numbers. Let I be the set of vertices in the Dynkin diagram of $G$ . In this paper I will work over base field $k=\mathbb{C}$ . We fix a Cartan subgroup $T$ of $G$ and a Borel subgroup $B\subset G$ . Denote by $N$ the unipotent radical of $B$ . Let $\varpi_{i},i\in I$ be the fundamental weights. Let $X_{*},X^{*}$ be the cocharacter, character lattice and $\langle\ ,\ \rangle$ be the pairing between them. Let W be the Weyl group. Let $e$ and $w_{0}$ be the unit and the longest element in $W$ . Let $\alpha_{i}$ and $\check{\alpha}_{i}$ be simple roots and coroots. Let $\Gamma=\{w\varpi_{i},w\in W,i\in I\}$ . $\Gamma$ is called the set of chamber weights.

Let $d$ be the formal disc and $d^{*}$ be the punctured formal disc. The ring of formal Taylor series is the ring of functions on the formal disc, $\mathcal{O}=\{\sum_{n\geq 0}a_{n}t^{n}\}$ . The ring of formal Laurent series is the ring of functions on the punctured formal disc, $\mathcal{K}=\{\sum_{n\geq{n_{0}}}a_{n}t^{n}\}$ .

For $X$ a variety, let Irr( $X$ ) be the set of irreducible components of $X$ .

2.2 MV cycles and polytopes

For a group $G$ , let $G_{\mathcal{K}}$ be the loop group of $G$ and $G_{\mathcal{O}}$ the disc group of $G$ . We define loop grassmannian $\mathcal{G}(G)$ as the left quotient $G_{\mathcal{O}}\setminus G_{\mathcal{K}}$ and view $\mathcal{G}(G)$ as an ind-scheme [BD91], [Zhu16]. An MV cycle is a certain finite dimentional subscheme in $\mathcal{G}(G)$ . For a cocharacter $\lambda\in X_{*}(T)$ , we denote the point it determines in $\mathcal{G}(G)$ by $L_{\lambda}$ . For $w\in W$ , let $N^{w}=wNw^{-1}$ . Define $S^{w}_{\lambda}=L_{\lambda}N^{w}_{\mathcal{K}}.$ This orbit is an ind-subscheme of $\mathcal{G}(G)$ and is called semi-infinite orbit since it is of infinite dimension and codimension in $\mathcal{G}(G)$ .
An irreducible component of $\overline{S_{0}^{e}\bigcap S^{w_{0}}_{\lambda}}$ is called an MV cycle of weight $\lambda$ . Kamnizter [Kam05] describes them as follows:

Theorem 1 ([Kam05]).

Given a collection of integers $(M_{\gamma})_{\gamma\in\Gamma}$ , if it satisfies edge inequalities, and certain tropical relations, put $\lambda_{w}=\sum_{i}M_{w\varpi_{i}}w\check{\alpha_{i}}.$
Then $\overline{\bigcap_{w\in W}S^{w}_{\lambda_{w}}}$ is an MV cycle, and each MV cycle arises from this way for the unique data $(M_{\gamma})$ .

The data $(M_{\gamma})_{\gamma\in\Gamma}$ determines a pseudo-Weyl polytope. It is called an MV polytope if the corresponding cycle $\overline{\bigcap_{w\in W}S^{w}_{\lambda_{w}}}$ is an MV cycle. MV polytopes are in bijection with MV cycles. Using this description, Kamnizter [Kam07] reconstruct the crystal structure for MV cycles.

Proposition 1 ([Kam07]).

MV polytopes have a crystal structure isomorphic to $B(\infty)$ .

2.3 Objects on the quiver side

Let $Q=\{I,E\}$ be a Dynkin quiver of type ADE, where I is the set of vertices and E is the set of edges. We double the edge set E by adding all the opposite edges. Let $E^{*}=\{a^{*}|a\in E\}$ where for $a:i\xrightarrow[]{}j,a^{*}=j\xrightarrow[]{}i$ , also we define $s(a)=i,t(a)=j$ . Define $\epsilon(a)=1$ when $a\in E$ , $\epsilon(a)=-1$ , when $a\in E^{*}$ . Let $H=E\bigsqcup E^{*}$ and $\overline{Q}=\{I,H\}$ . The preprojective algebra $\Pi$ of $Q$ is defined as quotient of the path algebra by a certain ideal:

\Pi_{Q}=k\overline{Q}/<\sum_{a\in H}\epsilon(a)aa^{*}>.

A $\Pi_{Q}-$ module is the data of an $I$ graded vector space $\bigoplus_{i\in I}M_{i}$ and linear maps $\phi_{a}:M_{s(a)}\xrightarrow[]{}M_{t(a)}$ for each $a\in H$ satisfying the preprojective relations $\sum_{a\in H,t(a)=i}\epsilon(a)\phi_{a}\phi_{a*}=0$ .

Given a dimension vector $d\in\mathbb{N}^{I}$ , define $\Lambda(d)$ to be the variety of all representations of $\Pi$ on M for $M_{i}=k^{d_{i}}$ .

Proposition 2 ([Lus90], [BK12]).

Irr $(\Lambda)$ has a crystal structure isomorphic to $B(\infty)$ .

2.4 A conjectural relation between MV cycles and modules over the preprojective algebra

Baumann and Kamnitzer found an isomorphism between the crystal structure of Irr $(\Lambda)$ and MV polytopes. For each $\gamma\in\Gamma$ , they define constructible funtion $D_{\gamma}:\Lambda(d)\xrightarrow[]{}\mathbb{Z}_{{\geq 0}}$ ²²2 $\Lambda(d)$ and $D_{\gamma}$ do not depend on the direction of the edges in E.. For any $M\in\Lambda(d)$ , the collection $(D_{\gamma})_{\gamma\in\Gamma}$ satisfies certain edge inequalities hence determines a polytope which we denote by $P(M)$ .

Theorem 2 ([BK12]).

When $M$ is generic, $P(M)$ is an MV-polytope and for $d=(d_{i})_{i\in I}$ this gives a map from Irr( $\Lambda(d))$ to the set of MV polytopes of weight $\sum_{i\in I}d_{i}\alpha_{i}$ . This map is a bijection compatible with the crystal structures.

We have MV-cycles (in bijection with MV-polytopes) as the geometric object on the loop Grassmannian side. In order to upgrade the relations geometrically, Kamnitzer-Knutson consider the quiver Grassmannian on the quiver side.

The quiver Grassmannian $Gr^{\Pi}(M)$ of a $\Pi$ -module $M$ is defined as the moduli of submodules of $M$ .

It is a subscheme of the moduli of $k$ -vector subspaces of $M$ which is product of usual grassmannian $\prod_{i\in I}Gr(M_{i})$ . Here we will only consider $Gr^{\Pi}(M)$ with its reduced structure, and actually just as a topological space. As the case of usual grassmannian, the quiver Grassmannian $Gr^{\Pi}(M)$ is disjoint union of Grassmannians of different dimension vectors. Denote $Gr_{e}^{\Pi}(M)$ by the moduli of submodule N of M of dimension vector $e$ , we have $Gr_{e}^{\Pi}(M)\subset\prod_{i\in I}Gr_{e_{i}}(M_{i})$ .

Given a module $M\in\Lambda(d)$ , form the subscheme³³3We will call it cycle in this paper. $X(M)$ = $\bigcap_{w\in W}\overline{S^{w}_{\lambda_{w}}}$ , where $\lambda_{w}=\sum_{i\in I}-D_{-w\varpi_{i}}(M)w\check{\alpha{{}_{i}}}$ . T acts on $S^{w}_{\lambda_{w}}$ by multiplication, hence it also acts on the closure and the intersection $X(M)$ .

Conjecture 1.

The ring of functions on the $T$ -fixed point subscheme of $X(M)$ is isomorphic to the cohomology ring of the quiver grassmannian of $M$

\mathcal{O}(X(M)^{T})\xrightarrow[\sim]{\Psi}H^{*}(Gr^{\Pi}(M)).

More precisely, $X(M)^{T}$ is disjoint union of finite schemes $X(M)^{T}_{\nu}$ supported at $L_{\nu}$ , $\nu\in X_{*}(T)$ and we can further identify two sides for each connnected component

\mathcal{O}(X(M)_{\nu}^{T})\xrightarrow[\sim]{\Psi}H^{*}(Gr_{e}^{\Pi}(M)),\text{ where }e_{i}=(\nu,\varpi_{i}).

Remark: we define $X(M)$ as a scheme theoretic intersection of closures while MV-cycles have been defined as varieties (closure of intersections). We notice that $X(M)$ may be reducible even when $P(M)$ is an MV-polytope. For an example, see the appendix⁴⁴4Not yet written, I will add this later on.. The former certainly contains the latter and a further hope is to relate the latter to some subvariety of the quiver Grassmannian.

3 The T fixed point subscheme of the cycle

We introduce some notation first. It is known that the $T$ -fixed point subscheme of the loop grassmannian of a reductive group $G$ is the loop grassmannian of the Cartan $T$ of $G$ , i.e., $\mathcal{G}(G)^{T}=\mathcal{G}(T)$ . We indentify $T$ with $I$ copies of the multiplicative group by $T\xrightarrow[\sim]{\prod\varpi_{i}}G_{m}^{I}$ and this gives $\mathcal{G}(T)\xrightarrow[\sim]{\prod\varpi_{i}}\mathcal{G}(G_{m})^{I}$ .

For $\mathcal{G}(G_{m})$ , we have

	$\displaystyle\mathcal{G}(G_{m})=G_{m}(\mathcal{O})\setminus G_{m}(\mathcal{K})$		(1)
	$\displaystyle=\{\text{unit}\in\mathcal{O}\}\setminus\{\text{unit}\in\mathcal{K}\}$		(2)
	$\displaystyle=t^{\mathbb{Z}}\cdot K_{-}$		(3)

where $K_{-}$ is called the negative congruence subgroup (of $G_{m}$ ). The $R$ -points of $K_{-}$ can be described as:

K_{-}(R)=\{a=(1+a_{1}t^{-1}+...+a_{m}t^{-m})|a_{i}\text{ is nilpotent in }R\}.

We define the degree function from $K_{-}(R)$ to $\mathbb{Z}_{\geq}$ : deg $(a)=m$ if $a_{m}\neq 0$ .

Then $(\bigcap\overline{S^{w}_{\lambda_{w}}})^{T}$ is a subscheme of $\mathcal{G}(G)^{T}\cong(t^{\mathbb{Z}}\cdot K_{-})^{|I|}$ .

Theorem 3.

Let $(\lambda_{w})_{w\in W}$ be a collection of cocharacters such that $\lambda_{v}\geq_{w}\lambda_{w}$ ⁵⁵5This notation is used in [Kam05], $\lambda_{v}\geq_{w}\lambda_{w}$ whenever $w^{-1}\lambda_{v}\geq w^{-1}\lambda_{w}$ . for all $w\in W$ in which case we know ([Kam05]) that $(\lambda_{w})_{w\in W}$ determines a pseudo-Weyl polytope. The integers $A_{w\varpi_{i}}$ are well defined by $A_{w\varpi_{i}}=(\lambda_{w},w\varpi_{i})$ . The R-points of $(\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$ is the subset of R-point of $(t^{\mathbb{Z}}\cdot K_{-})^{|I|}$ containing elements $(t^{(\nu,\varpi_{i})}a_{i})\in\prod(t^{\mathbb{N}}\cdot K_{-})^{|I|}|$ subject to the degree relations:

deg(\Pi_{i\in I}a_{i}^{(\gamma,\check{\alpha_{i}})})\leq-A_{\gamma}+\sum(\gamma,\nu)\text{ for all }\gamma\in\Gamma\}.

Proof.

We define loop grassmannian with a condition $Y$ and list the facts we need. For details, see [Mir17a]. Let $G$ acts on scheme $Y$ and $y$ be a point in $Y$ . Denote the stack quotient by $Y/G$ . Then $\mathcal{G}(G,Y)$ is the moduli of maps of pairs from $(d,d^{*})$ to $(Y/G,y)$ . When $Y$ is a point we recover $\mathcal{G}(G)$ . In general, $\mathcal{G}(G,Y)$ is the subfunctor of $\mathcal{G}(G)$ subject to a certain extension condition:

\mathcal{G}(G,Y)=G_{\mathcal{O}}\setminus\{g\in G_{\mathcal{K}}\ |\ d^{*}\xrightarrow[]{g}G\xrightarrow[]{o}Y\text{ extends to $d$}\},\text{ where }o(g)=gy.

We can realize semi-infinite orbits and their closures as follows:

•

$\mathcal{G}(G,G/N)=S_{0}$ , where G acts G/N by left multiplication.
•

$\mathcal{G}(G,(G/N)^{aff})=\overline{S_{0}}$ , where ”aff” means affinization.
•

$\mathcal{G}(G\times T,(G/N)^{aff})_{red}=\bigsqcup\overline{S_{\lambda}}$ , where ”red” means the reduced subscheme. Here T acts on $G/N$ by left multiplication with the inverse and this extends to an action on $(G/N)^{aff}$ .

•

\mathcal{G}(G\times\prod_{w\in W}T_{w},\prod_{w\in W}(G/N^{w})^{aff})=\bigsqcup_{(\lambda_{w})_{w\in W}}(\bigcap_{w\in W}\overline{S^{w}_{\lambda_{w}}})

, We denote a copy of T corresponding to $w\in W$ by $T_{w}$ .

A single cycle $\bigcap_{w\in W}\overline{S^{w}_{\lambda_{w}}}$ can be written as the fiber product:

\bigcap_{w\in W}\overline{S^{w}_{\lambda_{w}}}=\mathcal{G}(G\times\prod_{w\in W}T_{w},\prod(G/N^{w})^{aff})\times_{\mathcal{G}(\prod_{w\in W}T_{w})}(t^{\lambda_{w}})_{w\in W}.

In this fiber product, the morphism for the first factor is the second projection and the morphism for the second factor is the inclusion of the single point $t^{\underline{\lambda}}=(t^{\lambda_{w}})_{w\in W}$ .
For a reductive group G, we have $\mathcal{G}(G,Y)^{T}=\mathcal{G}(T,Y)$ , where T is the cartan of G.
So, the T fixed point subscheme is

(\bigcap_{w\in W}\overline{S^{w}_{\lambda_{w}}})^{T}=\mathcal{G}(T\times\prod_{w\in W}T_{w},\prod(G/N^{w})^{aff})\times_{\mathcal{G}(\prod_{w\in W}T_{w})}t^{\underline{\lambda}}.

In terms of the above extension condition, this fiber product is:

(\bigcap_{w\in W}\overline{S^{w}_{\lambda_{w}}})^{T}=T(\mathcal{O})\setminus\{g\in T_{\mathcal{K}},\text{ such that }d^{*}\xrightarrow[]{g,t^{\underline{\lambda}}}T\times T^{W}\xrightarrow[]{}\prod(G/N^{w})^{aff}\text{ extends to }d\}

This is the $T(\mathcal{O})$ quotient of the set of all $g\in T_{\mathcal{K}}$ , such that

d^{*}\xrightarrow[]{g,t^{\lambda_{w}}}T\times T_{w}\xrightarrow[]{}(G/N^{w})^{aff}\text{ extends to }d\text{ for all }w\in W.

For $\gamma\in W\cdot\varpi_{i}\subset\Gamma$ , we fix weight vectors $v_{\gamma}\text{ in the weight space}\ (V_{\varpi_{i}})_{\gamma}$ of $V_{\varpi_{i}}$ . For each $w\in W$ , we embed $G/N^{w}\text{ into }\bigoplus_{i\in I}V_{\varpi_{i}}\text{ by }g\mapsto(g\cdot v_{w\varpi_{i}})_{i\in I}$ . Under this embedding, $(G/N^{w})^{aff}$ is a closed subscheme in $\bigoplus_{i\in I}V_{\varpi_{i}}$ .

For $g\in T_{\mathcal{K}}$ , $w\in W$ , the composition $y_{w}(g)$ of the map :

d^{*}\xrightarrow[]{g,t^{\lambda_{w}}}T\times T_{w}\xrightarrow[]{}G/N^{w}\hookrightarrow\bigoplus V_{\varpi_{i}}

y_{w}(g)=(g\cdot(t^{\lambda_{w}})^{-1})\sum_{i\in I}v_{w\varpi_{i}}=\sum_{i\in I}(w\varpi_{i}(g\cdot t^{-\lambda_{w}}))v_{w\varpi_{i}}.

This map extends to $d$ when for each $i\in I$ , the coefficient of $v_{w\varpi_{i}}$ is in $\mathcal{O}$ . The coefficient of $v_{w\varpi_{i}}$ is

	$\displaystyle w\varpi_{i}(g\cdot t^{-\lambda_{w}})=w\varpi_{i}(g)\cdot w\varpi_{i}(t^{-\lambda_{w}})=w\varpi_{i}(g)\cdot t^{-(w\varpi_{i},\ \lambda_{w})}$
	$\displaystyle=w\varpi_{i}(g)t^{-A_{w\varpi_{i}}}=\gamma(g)z^{-A_{\gamma}}\text{ where }\gamma=w\varpi_{i}.$

It follows that

(\bigcap_{w\in W}\overline{S^{w}_{\lambda_{w}}})^{T}=T(\mathcal{O})\setminus\{g\in T(\mathcal{K});\gamma(g)t^{-A_{\gamma}}\in\mathcal{O}\text{ for all }\gamma\in\Gamma\}.

and the description of the R-points of $(\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$ in the theorem follows when we identify $\mathcal{G}(T)\xrightarrow[]{\prod\varpi_{i}}\mathcal{G}(G_{m})^{I}=(t^{\mathbb{Z}}\cdot K_{-})^{I}$ .

∎

3.1 Ring of functions on $(\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$

For an R-point $(t^{(\nu,\varpi_{i})}a_{i})_{i\in I}$ of $(\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$ , let us write $a_{i}=1+a_{i1}t^{-1}+\cdots+a_{im}t^{-m}$ . When $\gamma=\varpi_{i}$ , the degree inequality is deg $(a_{i})\leq(\varpi_{i},\nu)-A_{\varpi_{i}}$ . We can take the coefficients $a_{ij}$ to be the coordinate functions on $(\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$ . Since deg $(a_{i})\leq(\varpi_{i},\nu)-A_{\varpi_{i}}$ , there are finitely many $a_{ij}$ s which generate the ring of functions on $\mathcal{O}((\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T})$ .
When we take inverse of $a_{i}$ , it is computed in $K_{-}$ as $a_{i}^{-1}=1+\sum_{s\geq 0}(-1)^{i}(a_{i1}t^{-1}+\cdots+a_{im}t^{-m})^{s}$ and then expands in the form $\sum_{i}b_{ik}t^{-k}$ , where $b_{ik}$ is the coefficient of $t^{-k}$ in $a_{i}^{-1}$ .

deg(\Pi_{i\in I}a_{i}^{(\gamma,\check{\alpha_{i}})})\leq-A_{\gamma}+\sum(\gamma,\nu)\text{ for all }\gamma\in\Gamma.

is equivalent to the condition that the coefficient of the term $t^{-1}$ to the power $-A_{\gamma}+\sum(\gamma,\nu)+1$ in $(\Pi_{i\in I}a_{i}^{(\gamma,\check{\alpha_{i}})})$ is 0. These coefficients are polynomials of $a_{ij}$ s. Set $b_{i}=1+\sum_{k}b_{ik}t^{-k}=a_{i}^{-1}$ , add $b_{ij}$ ’s as generaters and also add the relations $a_{i}b_{i}=1$ for $i\in I$ which eliminate all $b_{ij}$ ’s. For $\gamma\in\gamma$ , let $\gamma_{i}=(\gamma,\check{\alpha_{i}})$ . Denote by $I_{\gamma}^{+}$ the subset of I containing all i such that $\gamma_{i}$ is positive and by $I_{\gamma}^{-}$ containing all i $\gamma_{i}$ is negative. Set $\gamma^{+}_{i}=\gamma_{i}$ when $\gamma_{i}$ is positive and $\gamma^{-}_{i}=-\gamma_{i}$ when $\gamma_{i}$ negative.

Corollary 1.

The ring of functions on $\mathcal{O}((\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T})$ is generated by $a_{ij}$ ’s and $b_{ik}$ ’s, for $i\in I$ . The relations are degree conditions:

deg(\prod_{i\in I_{\gamma}^{+}}a_{i}^{\gamma{{}_{i}}^{+}}\prod_{i\in I_{\gamma}^{-}}b_{i}^{\gamma{{}_{i}}^{-}})\leq(\gamma,\nu)-A_{\gamma}

for each $\gamma\in\Gamma$ and conditions $a_{i}b_{i}=1$ for each i in I.

4 Construction of the map $\Psi$ from functions to cohomology

4.1 Map $\Psi$

For $M\in\Lambda(d)$ , to apply corollary 1 to $X(M)$ , we set $A_{\gamma}=-D_{-\gamma}(M)$ . Then

\mathcal{O}(X(M)_{\nu}^{T})=k[a_{ij},b_{ik}]/I(M)

where $I(M)$ is the ideal generated by the degree conditions:

deg(\prod_{i\in I_{\gamma}^{+}}(a_{i})^{\gamma_{i}^{+}}\prod_{i\in I_{\gamma}^{-}}(b_{i})^{\gamma_{i}^{-}})\leq(\gamma,\nu)+D_{-\gamma}(M)

for each $\gamma\in\Gamma$ and the conditions $a_{i}b_{i}=1$ for each i in I.

The conjecture $\mathcal{O}(X(M)_{\nu}^{T})\cong H^{*}(Gr_{e}^{\Pi}(M))$ , where $e_{i}=(\nu,\varpi_{i})$ , is now equivalent to

k[a_{ij},b_{ik}]/I(M)\cong H^{*}(Gr_{e}^{\Pi}(M)).

The quiver Grassmannian $Gr_{e}^{\Pi}(M))$ is a subvareity of $\prod_{i\in I}Gr_{e_{i}}(M_{i})$ and we have on each $Gr_{e_{i}}(M_{i})$ the tautological subbundle $S_{i}$ and quotient bundle $Q_{i}$ . We pull back $S_{i}$ and $Q_{i}$ to $\prod_{i\in I}Gr_{e_{i}}(M_{i})$ and denote their restrictions on $Gr_{e}^{\Pi}(M))$ still by $S_{i}$ and $Q_{i}$ by abusing notion. For a rank n bundle E, denote the Chern class by $c(E)$ and the $i^{t}h$ Chern class $c_{i}(E)$ , where $c(E)=1+c_{1}(E)+\cdots+c_{n}(E)$ . We want to define the map

\Psi:\mathcal{O}(X(M)_{\nu}^{T})\xrightarrow[]{}H^{*}(Gr_{e}^{\Pi}(M)),\text{ where }e_{i}=(\nu,\varpi_{i}),

by mapping the generators $a_{ij}$ to $c_{j}(S_{i})$ and $b_{ij}$ to $c_{j}(Q_{i})$ .

Theorem 4.

The map $\Psi$ described above is well defined.

4.2 Two lemmas

For the proof, we need two lemmas. Lemma 1 is the special case of theorem 4 when $Q$ is the quiver $1\xrightarrow[]{}2$ and $M$ is a $kQ$ -module.

Lemma 1.

Let $Q$ be the quiver $1\xrightarrow[]{}2$ and $M$ be $\mathbb{C}^{d_{1}}\xrightarrow[]{\phi}\mathbb{C}^{d_{2}}$ .On $X=Gr^{\Pi}_{e}(M)$ , we have $c_{i}(S_{2}\oplus Q_{1})=0$ when $i>e_{2}-e_{1}+\text{dim}(ker\phi)$ .

Let $\phi_{ij}:M_{i}\xrightarrow[]{}M_{j}$ be the composition of $\phi_{a}$ where a travels over the unique no going-back path which links i and j. Let $M_{\gamma}=\oplus_{i\in I_{\gamma}^{-}}M_{i}^{\gamma^{-}}\xrightarrow[]{\phi_{\gamma}=\oplus\phi_{ij}}\oplus_{i\in I_{\gamma}^{+}}M_{i}^{\gamma^{+}}$ be the module over $k(1\xrightarrow[]{}2)$ .

Lemma 2.

For a $\Pi$ -module $M$ and any chamber weight $\gamma$ , we have

dim(ker\phi_{\gamma})=D_{-\gamma}(M).

Lemma 2 is a property of $D_{\gamma}$ and will be proved in the appendix.

4.3 Proof of theorem 4 from lemmas in $\mathsection$ 4.2

Proof of theorem 4.

We prove the theorem can be reduced to lemma 1.

For each $\gamma\in\Gamma$ ,we have to prove the degree inequilities carry over to Chern classes:

\Psi(\prod_{i\in I_{\gamma}^{+}}t_{i}^{\gamma_{i}^{+}}\prod_{i\in I_{\gamma}^{-}}s_{i}^{\gamma_{i}^{-}})=\prod_{i\in I_{\gamma}^{+}}c(S_{i})^{\gamma_{i}^{+}}\prod_{i\in I_{\gamma}^{-}}c(Q_{i})^{\gamma_{i}^{-}}\leq D_{w_{0}\gamma}(M)+(\nu,\gamma).

Define a map $\Phi$ from $Gr^{\Pi}(M)$ to $Gr^{k(1\xrightarrow[]{}2)}(M_{\gamma})$ : for $N\in Gr^{\Pi}(M)$ , $\Phi(N)=\oplus_{i\in I_{\gamma}^{-}}N_{i}\xrightarrow[]{\phi_{\gamma}}\oplus_{i\in I_{\gamma}^{+}}N_{i}$ . We have

\Phi^{*}(c(S_{2})c(Q_{1}))=c(\Phi^{*}(S_{2}))c(\Phi^{*}(Q_{1}))=c(\oplus_{i\in I_{\gamma}^{+}}S_{i}^{\gamma_{i}^{+}})c(\oplus_{i\in I_{\gamma}^{-}}Q_{i}^{\gamma_{i}^{-}})

=\prod_{i\in I_{\gamma}^{+}}c(S_{i})^{\gamma_{i}^{+}}\prod_{i\in I_{\gamma}^{-}}c(Q_{i})^{\gamma_{i}^{-}}.

Apply lemma1 to $M_{\gamma}$ we have

deg(c(Q_{1})c(S_{2}))\leq dimker(\phi_{\gamma})+\sum_{i\in I_{\gamma}^{+}}\gamma_{i}e_{i}-\sum_{i\in I_{\gamma}^{-}}\gamma_{i}e_{i}

=dimker(\phi_{\gamma})+\sum_{i\in I}\gamma_{i}e_{i}=dimker(\phi_{\gamma})+(\gamma,\nu).

Then the theorem follows by lemma 2. ∎

Chern class vanishes in certain degree when the bundle contains a trivial bundle of certain degree but the desired trivial bundle in $Q_{1}\oplus S_{2}$ does not exist. The idea is to pass to T-equivariant cohomology. Over $X^{T}$ which is just a union of isolated points we will decompose $Q_{1}\oplus S_{2}$ into the sum of the other two bundles $E_{1}$ and $E_{2}$ pointwisely, where $E_{1}$ will play the role of trivial bundle. Although there is no bundle over X whose restriction is $E_{2}$ , there exist T-equivariant cohomology class in $H_{T}^{*}(X)$ whose restriction on $X^{T}$ is the T-equivariant Chern class of $E_{2}$ .

4.4 Recollection of GKM theory

We first recall some facts in T-equivariant cohomoloy theory.

We follow the paper [Tym05]. Denote a n-dimensional torus by $T$ , topologically $T$ is homotopic to $(S^{1})^{n}$ . Take $ET$ to be a contractible space with a free $T$ -action. Define $BT$ to be the quotient $ET/T$ . The diagonal action of $T$ on $X\times ET$ is free, since the action on $ET$ is free. Define $X\times_{T}ET$ to be the quotient $(X\times ET)/T$ . We define the equivariant cohomology of $X$ to be

H^{*}_{T}(X)=H^{*}(X\times_{T}ET).

When $X$ is a point and $T=G_{m}$ ,

H^{*}_{T}(X)=H^{*}(\textup{pt}\times_{T}ET)=H^{*}(ET/T)=H^{*}(BT)=H^{*}(\mathbb{CP}^{\infty})\cong k[t].

When $T=(S^{1})^{n}$ ,

\displaystyle H^{*}(pt)=k[t_{1},\cdots,t_{n}]\cong S(\mathfrak{t}^{*}).

(4)

So we can identify any class in $H^{*}(pt)$ as a function on the lie algebra $\mathfrak{t}$ of $T$ . The map $X\xrightarrow[]{}pt$ allows us to pull back each class in $H^{*}_{T}(\textup{pt})$ to $H^{*}_{T}(X)$ , so $H^{*}_{T}(X)$ is a module over $H^{*}_{T}(\textup{pt})$ .

Fix a projective variety $X$ with an action of $T$ . We say that $X$ is equivariantly formal with respect to this $T$ -action if $E^{2}=E^{\infty}$ in the spectral sequence associated to the fibration $X\times_{T}ET\longrightarrow BT$ .

When $X$ is equivariantly formal with respect to $T$ , the ordinary cohomology of $X$ can be reconstructed from its equivariant cohomology. Fix an inclusion map $X\xrightarrow[]{}X\times_{T}ET$ , we have the pull back map of cohomologies: $H^{*}(X\times_{T}ET\xrightarrow[]{i}H^{*}(X)$ . The kernel of $i$ is $\sum_{s=1}^{n}t_{s}\cdot H^{*}_{T}(X)$ , where $t_{s}$ is the generator of $H^{*}_{T}(pt)$ (see (4)) and we view it as an element in $H^{*}_{T}(X)$ by pulling back the map $X\xrightarrow[]{}pt$ . Also $i$ is surjective so $H^{*}(X)=H^{*}_{T}(X)/ker(i)$ .

If in addition $X$ has finitely many fixed points and finitely many one-dimensional orbits, Goresky, Kottwitz, and MacPherson show that the combinatorial data encoded in the graph of fixed points and one-dimensional orbits of $T$ in $X$ implies a particular algebraic characterization of $H^{*}_{T}(X)$ .

Theorem 5 (GKM, see [Tym05], [GKM97]).

Let $X$ be an algebraic variety with a $T$ -action with respect to which $X$ is equivariantly formal, and which has finitely many fixed points and finitely many one-dimensional orbits. Denote the one-dimensional orbits $O_{1}$ , $\ldots$ , $O_{m}$ . For each $i$ , denote the the $T$ -fixed points of $O_{i}$ by $N_{i}$ and $S_{i}$ and denote the stabilizer of a point in $O_{i}$ by $T_{i}$ . Then the map $H^{*}_{T}(X)\xrightarrow[]{l}H^{*}_{T}(X^{T})=\oplus_{p_{i}\in X^{T}}H^{*}_{T}(p_{i})$ is injective and its image is

\left\{f=(f_{p_{1}},\ldots,f_{p_{m}})\in\bigoplus_{\textup{fixed pts}}S(t^{*}):f_{N_{i}}|_{\mathfrak{t}_{i}}=f_{S_{i}}|_{\mathfrak{t}_{i}}\textup{ for each }i=1,\ldots,m\right\}.

Here $\mathfrak{t}_{i}$ is the lie algebra of $T_{i}$ .

4.5 Affine paving of $Gr^{\Pi}_{e}(M)$ when $M$ is a representation of $Q$ of type A

Definition 1 ([Tym07] 2.2).

We say a space $X$ is paved by affines if $X$ has an order partition into disjoint $X_{1},X_{2},\cdots$ such that each finite union $\bigcup_{i=1}^{j}X_{i}$ is closed in X and each $X_{i}$ is an affine space.

A space with an affine paving has odd cohomology vanishing.

Proposition 3 ([Tym07], 2.3).

Let $X=\bigcup X_{i}$ be a paving by a finite number of affines with each $X_{i}$ homeomorphic to $\mathbb{C}^{d_{i}}$ . The cohomology groups of $X$ are given by $H^{2k}(X)=\bigoplus_{\{i\in I\ |\ d_{i}=k\}}\mathbb{Z}$ .

The main observation is the following lemma.

Lemma 3.

Let $M$ be a representation of Q, where Q is of type A with all edges in E pointing to the right. Then the quiver Grassmannian $Gr^{\Pi}_{e}(M)$ is paved by affines for any dimension vector e.

We need a sublemma first.

Sublemma 1.

Suppose X is paved by $X_{i}$ ’s. Let $Y\subset X$ be a subspace. if for each i, $Y_{i}=X_{i}\bigcap Y$ is $\emptyset$ or affine then $Y=\bigcup Y_{i}$ is an affine paving.

Proof.

$\bigcup_{i\leq j}Y_{i}=\bigcup_{i\leq j}(X_{i}\bigcap Y)=(\bigcup_{i\leq j}X_{i})\bigcap Y$ is closed in $Y$ since $\bigcup_{i\leq j}X_{i}$ is closed in $X$ . ∎

Proof of lemma 3.

Let $V=\oplus M_{i}$ be the underlying vector space and $\phi=\oplus_{a\in H}\phi_{a}$ be the nilpotent operator on $V$ . We adopt the notations in [Shi85]. Let $n=dimV$ and $d=\sum e_{i}$ . Let $C^{\phi}_{\alpha}=\{(v_{1},...v_{d})\in C_{\alpha},\phi v_{i}=v_{j}$ if $\alpha$ contains $\alpha_{i}$ $\alpha_{j}$ $,(1\leq i<j<d)\}$ . We know that $Gr_{d}(k^{n})^{\phi}=\bigsqcup C_{\alpha}^{\phi}$ . We want to show $S^{\phi}_{\alpha}\bigcap Gr^{\Pi}_{e}(M)$ is affine. Take $x\in S^{\phi}_{\alpha}\bigcap Gr^{\Pi}_{e}(M)$ , from 1.10 in [Shi85], $x=v_{1}\wedge...\wedge v_{d}$ where $x=\wedge(\phi^{h}w_{i}:i^{\prime}s$ are the initial numbers of $\alpha$ and $\phi^{h}w_{i}\neq 0).$
We now show that for $x=v_{1}\wedge\cdots\wedge v_{d}\in Gr^{\Pi}_{e}(M)$ , where $v_{i}=e_{i}+\sum_{j\neq i}x_{i}(j)e_{j}$ , if $e_{i}\in M_{t}$ , we have $v_{i}\in M_{t}$ . Conversely, if for each i, there exists t such that $v_{i}\in M_{t}$ , $x\in Gr^{\Pi}_{e}(M)$ .Denote this t determined uniquely by i as t(i).
Since $span(v_{1},...,v_{d})$ is a direct sum of some $N_{i}\subset M_{i}$ , we have $Pr_{t}(v_{i})\in span(v_{1},...,v_{d})$ , where $Pr_{t}$ is the projection from V to $M_{t}$ , and so $Pr_{t}(v_{i})=\sum a_{p}v_{p}$ . Comparing the coefficient of $e_{p}$ , by the definition of $C_{\alpha}$ , we have $a_{p}=0$ for $p\neq i$ . So we have $Pr_{t}(v_{i})=v_{i}$ , which implies $v_{i}\in M_{t}$ .
Note that $\{v_{1},\cdots,v_{d}\}$ is determined by $w_{i}$ where i is an initial number(and vice versa).
We have $w_{i}\in M_{t(i)}$ . Denote $l(i)$ be the number on the left of i in the d-tableaus. If i is the leftmost, set $l(i)$ to be $\emptyset$ , and set $e_{\emptyset}=0$ . Write $w_{i}=e_{i}+\sum x_{ij}e_{j}$ , where $e_{j}\in M_{t_{i}}$ , we have $\phi^{r}(w_{i})=e_{l^{r}(i)}+\sum x_{ij}e_{l^{r}(j)}$ . Since M is kQ-module, we have $l^{r}(i)=l^{r}(j)$ hence $v_{i_{r}}\in M_{t(i_{r})}$ and $x\in Gr^{\Pi}_{e}(M)$ . So we have $S^{\phi}_{\alpha}\bigcap Gr^{\Pi}_{e}(M)$ is affine. Apply lemma 3, we are done.

∎

4.6 Proof of lemma 1

Proof.

For $M$ given by $\mathbb{C}^{d_{1}}\xrightarrow[]{\phi}\mathbb{C}^{d_{2}}$ and a choice of $e=(e_{1},e_{2})$ , denote $X=Gr^{\Pi}_{e}(M)$ . First, we define a torus action on $X$ . Let $I=ker\phi$ . Choose a basis $e_{1},e_{2},\cdots,e_{s}$ of $I$ and extend it to a basis $e_{1},\cdots,e_{s},e_{s+1},\cdots,e_{t}$ of $M_{1}$ . Let $J$ be span $\{e_{s+1},\cdots,e_{t})\}$ so the image of $J$ is span $\{f_{s+1},\cdots,f_{t}\}$ . We extend the basis $\{f_{i}=\phi(e_{i})\}$ of the image of $J$ to a basis ( $f_{s+1},\cdots,f_{t},f_{t+1},\cdots,f_{r})$ of $M_{2}$ . Let K=span $\{f_{t+1},...f_{r}\}$ . we have $M_{1}=I\oplus J$ and $M_{2}=\phi(J)\oplus K$ .

Let $\mathcal{I}=\{1,\cdots,s\}$ , $\mathcal{J}=\{s+1,\cdots,t\}$ and $\mathcal{L}=\{t+1,\cdots,r\}$ . Let tori $T_{I}=G_{m}^{\mathcal{I}},T_{J}=G_{m}^{\mathcal{J}},T_{L}=G_{m}^{\mathcal{L}}$ act on $I,J\cong\phi(J)$ , $K$ by multiplication compotentwisely (For instance, $T_{I}$ acts on I by $(t_{1},\cdots,t_{s})\sum a_{i}e_{i}=\sum a_{i}t_{i}e_{i}$ and on $J,K$ trivially). Hence they act on $M_{1}=I\oplus J$ and $M_{2}=\phi(J)\oplus K$ . This induces an action of $T=T_{I}\times T_{J}\times T_{K}$ on $Gr^{\Pi}_{e}(M)$ . By lemma 3, $Gr^{\Pi}_{e}(M)$ is paved by affines so by proposition 3 it has odd cohomology vanishing therefore the spectral sequence associated to the fibration $X\times_{T}ET\longrightarrow BT$ converges at $E^{2}$ and X is equivariantly formal.

Denote by $f$ the forgetful map $H^{*}_{T}(X)\xrightarrow[]{f}H^{*}(X)$ . From $\mathsection 4.4$ we have $ker(f)=\sum_{1}^{dimT}t_{s}H^{*}_{T}(X)$ . Since $c^{i}(S_{2}\oplus Q_{1})=f(c^{i}_{T}(S_{2}\oplus Q_{1}))$ , it suffices to prove $c^{i}_{T}(S_{2}\oplus Q_{1}))\in ker(f)$ when $i>e_{2}-e_{1}+dimI$ .

To use GKM theorem, we need to know the one dimensional orbits and T-fixed points of X.

First, we see what $X^{T}$ is. For a point $p=(V_{1},V_{2})$ in $X$ , in order to be fixed by $T$ , $V_{1}$ and $V_{2}$ need to be spanned by some of basis vectors $e_{i}$ and $f_{i}$ . For a subset $S$ of $\mathcal{I}$ (resp. $\mathcal{J}$ ), we denote by $e_{S}$ (resp. $f_{S}$ ) the span $\{e_{i}|i\in S\}$ (resp. span $\{f_{i}|i\in S\}$ ). The T-fixed points in $X$ consist of all $V=(V_{1},V_{2})$ , such that $V_{1}=e_{A\bigcup B},V_{2}=f_{C\bigcup D}$ , for some $A\subset I,B\subset C\subset J$ and $D\subset K$ .

For any point $p=(V_{1},V_{2})$ in $X^{T}$ , let $V_{1}=e_{A\bigcup B},V_{2}=f_{C\bigcup D}$ . Over $p$ , $Q_{1}=(I\oplus J)/e_{A\bigcup B}$ is isomorphic to $e_{(\mathcal{I}\setminus A)\oplus(\mathcal{J}\setminus B)}$ (The restriction of a $T$ -equivariant bundle to a T-fixed point is just a $T$ -module). So over $X^{T}$ , we can decompose $S_{2}\oplus Q_{1}$ as follows:

S_{2}\oplus Q_{1}\cong e_{(\mathcal{I}\setminus A)\oplus(\mathcal{J}\setminus B)}\oplus f_{(C\bigcup D)}=(e_{(\mathcal{I}\setminus A)\oplus(C\setminus B)}\oplus f_{D})\oplus(e_{\mathcal{J}\setminus C}\oplus f_{C}).

Denote the bundle over $X^{T}$ whose fiber over each point p is $e_{(\mathcal{I}\setminus A)\oplus(C\setminus B)}\oplus f_{D})$ by $E_{1}$ and the bundle over $X^{T}$ whose fiber over p is $e_{(\mathcal{J}\setminus C)}\oplus f_{C}$ by $E_{2}$ .

We now use localization. Denote by $l$ the map $H^{*}_{T}(X)\xrightarrow[]{l}H^{*}_{T}(X^{T})=\oplus_{p\in X^{T}}H^{*}(p)$ . From GKM theory $l$ is injective, so the condition $c^{i}_{T}(S_{2}\oplus Q_{1}))\in ker(f)$ is equivalent to $l(c^{i}_{T}(S_{2}\oplus Q_{1}))\in l(ker(f))$ . We have

\displaystyle l(ker(f))=l(\sum_{s=1}^{dimT}t_{s}H^{*}_{T}(X))=\sum_{s=1}^{dimT}\underbrace{(t_{s},\cdots,t_{s})}_{\text{the number of T-fixed points in X}}l(H^{*}_{T}(X)).

(5)

By functorality of Chern class, $l(c^{i}_{T}(S_{2}\oplus Q_{1}))=c^{i}_{T}(S_{2}|_{X^{T}}\oplus Q_{1}|_{X^{T}}))$ .
We compute the $T$ -equivariant Chern class over $X^{T}$ . For⁶⁶6We always denote S and Q but indicate over which space we are considering. each p,

c_{T}^{p}(S_{2}\oplus Q_{1})=c_{T}^{p}(E_{2}\oplus E_{1})=\sum_{i}c_{T}^{p-i}(E_{1})c_{T}^{i}(E_{2})=\sum_{i\geq 1}c_{T}^{p-i}(E_{1})c_{T}^{i}(E_{2}).

The last equality holds since $c_{T}^{p}(E_{2})=0$ when $p>dimE_{2}=dimI+e_{2}-e_{1}$ . Now to show $c_{T}^{p}(S_{2}\oplus Q_{1})\in l(ker(f))$ , It suffices to show that $c_{T}^{p-i}(E_{1})c_{T}^{i}(E_{2})\in l(ker(f))$ , for any $i$ . The action of T on $E_{2}$ is actually the same on each T-fixed point. And at each point, $c_{T}^{i}(E_{2})$ is the $i^{th}$ elementary symmetric polynomial of $t_{s},1\leq s\leq dimT$ . So by (5), it suffices to show that $c_{T}^{i}(E_{1})\in l(H_{T}^{*}(X))$ .

Now we will see what 1-dimensional orbits are. Take an orbit $O_{i}$ , in order to be 1 dimensional its closure must contain two fixed points. Let $\overline{O_{i}}=O_{i}\bigcup\{N_{i}\}\bigcup\{S_{i}\}$ , where $N_{i}=(e_{A\bigcup B},f_{C\bigcup D})$ and $S_{i}=(e_{A^{\prime}\bigcup B^{\prime}},f_{C^{\prime}\bigcup D^{\prime}})$ are the fixed points. $O_{i}$ is one dimensional whenever either $A\bigcup B$ and $A^{\prime}\bigcup B^{\prime}$ differ by one element with $C\bigcup D=C^{\prime}\bigcup D^{\prime}$ or $C\bigcup D$ and $C^{\prime}\bigcup D^{\prime}$ differ by one element with $A\bigcup B=A^{\prime}\bigcup B^{\prime}$ . In the first case, we have some $s\in A\bigcup B$ and $s^{\prime}\in A^{\prime}\bigcup B^{\prime}$ , such that $A\bigcup B\setminus s=A^{\prime}\bigcup B^{\prime}\setminus s^{\prime}$ .

Notice that the annihilator for the lie algebra $\mathfrak{t}_{i}$ in $S(t^{*})$ is generated by $t_{s}-t_{s^{\prime}}$ , so by theorem 5, the condition along $O_{i}$ for an element $h\in H^{*}_{T}(X^{T})$ to be in $im(l)$ is

(t_{s}-t_{s^{\prime}})\ |\ (h_{N_{i}}-h_{S_{i}}).

But we have

c_{T}(E_{1})|_{N_{i}}-c_{T}(E_{1})|_{S_{i}}=\\ (1+t_{s{\prime}})\prod_{i\in\mathcal{I}\bigcup C\setminus(A\bigcup B)\setminus\{s^{\prime}\}}(1+t_{i})-(1+t_{s})\prod_{i\in\mathcal{I}\bigcup C\setminus(A^{\prime}\bigcup B^{\prime})\setminus\{s\}}(1+t_{i}).

Note that $\mathcal{I}\bigcup C\setminus(A\bigcup B)\setminus\{s^{\prime}\}=\mathcal{I}\bigcup C\setminus(A^{\prime}\bigcup B^{\prime})\setminus\{s\}$ , so $t_{s}-t_{s^{\prime}}$ divides $c_{T}(E_{1})|_{N_{i}}-c_{T}(E_{1})|_{S_{i}}$ . We conclude that $c_{T}^{i}(E_{1})\in l(H_{T}^{*}(X))$ .
The other case is similar.

∎

5 Proof of isomorphism when $M$ is a representation of $Q$ of type A

We first prove that $\Psi$ is surjective.

Lemma 4.

(a) Denote $Y=\prod Gr_{e_{i}}(k^{d_{i}})$ and $X=Gr^{\Pi}_{e}(M)$ . Then $Y\setminus X$ is paved by affines.
(b) $\Psi$ is surjective.

Proof.

(a). Let a be the number where the Young diagram of $\phi_{Y}$ has $a^{th}$ row as the first row from the bottom that does not have one block. For example, in the left diagram, a=4.

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Define $\phi^{\prime}$ be the operator of $V$ that corresponds to the diagram by moving the left most block A of the $a^{th}$ row to the bottom in the diagram of $\phi_{Y}$ . Let $M^{\prime}$ be the corresponding module and $X^{\prime}$ be $Gr_{e}{M^{\prime}}$ .
We claim that $X^{\prime}\setminus X$ is paved by affines.
By lemma 4, we have $X=\bigsqcup_{\alpha\in I}C_{\alpha}$ , where $I^{\prime}$ is the set of all semi-standard young tableau in $\lambda$ . Also we have $Y^{\prime}=\bigsqcup_{\alpha\in I^{\prime}}C^{\prime}_{\alpha}$ , where $I$ is the set of all semi-standard young tableau in $\lambda^{\prime}$ . If $\alpha$ contains block A, $\alpha$ is s.s in $\lambda$ implies $\alpha^{\prime}$ is s.s in $\lambda^{\prime}$ . If $\alpha$ does not contain block A , $\alpha$ also does not contain any block in that row, so $\alpha$ is still s.s in $\lambda^{\prime}$ . So $I\subset I^{\prime}$ .
For $\alpha$ that contains block A, there are two types. Let E be the set of $\alpha$ that contains block A and some other block in the row of A. Let F be the set of $\alpha$ that contains block A but no other block in the row of A. Let G be the set of $\alpha$ that does not contain block A. So we have $I=E\bigsqcup F\bigsqcup G=F\bigsqcup(E\bigsqcup G)$ .
Take $\alpha\in F$ , in $\lambda$ , the block A in $\alpha$ is not initial so the vector indexed by A is determined by the initial vector. In $\lambda^{\prime}$ , A is the last block so the vector indexed by A is the basis vector indexed by block A. In both case the vector indexed by A has been determined, so $C_{\alpha}=C_{\alpha}^{\prime}$ when $\alpha\in F.$

For $\alpha\in E\bigcup G$ , let $s(\alpha)$ be the tableau of the same relative position in $\lambda^{\prime}$ as $\alpha$ in $\lambda$ . Then $C_{\alpha}=C^{\prime}_{s(\alpha)}$ . Since s is a bijection between $E\bigsqcup G$ and $E^{\prime}\bigsqcup G^{\prime}$ we have $\bigsqcup_{\alpha\in E\bigsqcup G}C_{\alpha}=\bigsqcup_{\alpha\in E^{\prime}\bigsqcup G^{\prime}}C^{\prime}_{\alpha}$ .
Then we have $X^{\prime}\setminus X=\bigsqcup_{\alpha\in I^{\prime}\setminus I}C^{\prime}_{\alpha}$ is paved by affines. We can do this procedure step by step until $X^{\prime}$ becomes Y, so we are done.

(b). By lemma 1, X is paved. With part(a) , we have the homology map from X to Y is injective hence $\Psi$ is surjective (see 2.2 in[Tym07]). ∎

We want to prove the two sides of $\Psi$ have the same dimension as k-vector spaces and actually we will prove it for a more general setting.

Definition 2.

For a $\Pi$ -mod $M$ Let $V=\oplus M_{i}$ be the underlying vector space and $\phi=\oplus_{a\in H}\phi_{a}$ be the nilpotent operator on $V$ . We say $M$ is $I$ -compatible if there is a Jordan basis $\{v_{ij}\}$ of V such that each $v_{ij}$ is contained in some $M_{r}$ .

Definition 3.

For a $\Pi$ -module $M$ , if the Young diagram of the associated operator $\phi$ has one raw, we call $M$ one direction module.

Proposition 4.

If $M$ is I-compatible, it is a direct sum of one direction module with multiplicities.

Lemma 5.

If M is I-compatible, the dimension of two sides of $\Psi$ have the inequality: dim $k[a_{ij},b_{ik}]/I(M,e))\leq\chi(Gr^{\Pi}_{e}(M))$ , where $\chi$ is the Euler character. We denote the ring on the left by $R(M,e)$ .

First we state a lemma due to Caldero and Chapoton.

Lemma 6 (see prop 3.6 in [CC04]).

For $\Pi$ -module $M,N$ , we have

\chi(Gr_{g}(M\oplus N)=\sum_{d+e=g}\chi(Gr_{d}(M))\chi(Gr_{e}(N)).

The following two lemmas are proved after the proof of lemma 5.

Lemma 7.

$R(M,e)/<b_{1(d_{1}-e_{1})}>\cong R(M^{\prime},e)$ .

Lemma 8.

$b_{1(d_{1}-e_{1})}R(M,e)$ is a module over $R(M^{\prime\prime},e-\sum_{i\in I}\alpha_{i})$ .

Proof of lemma 5.

⁷⁷7The rest of this chapter is not well-written and should be revised.

We index the basis vector according to the Young diagram of $\phi$ as before but slightly different: $e_{ij}$ corresponds to the block of $i^{th}$ row (from up to down) and $j^{th}$ column (from right to left, which is the difference from before and this will cause the problem that two blocks in the same column but different row have different j but we will fix an i sooner so will not be of trouble). so $\phi(e_{ij})=e_{i(j-1)}$ .
The basis vector in $M_{1}$ appears in the first column or in the last column. If it lies in the last, we can take the dual to make it in the first. So, there exist i such that $e_{i1}\in M_{1}$ .
Let $\lambda^{\prime}$ be the young diagram removing the block of $e_{i1}$ from the original one and $M^{\prime}$ be the corresponding module. Let $\lambda^{\prime\prime}$ be the young diagram removing $i^{th}$ row and $M^{\prime\prime}$ be the corresponding module. Apply lemma 6, we have

\chi(Gr^{\Pi}_{e}(M))=\chi(Gr_{e}(M^{\prime}))+\chi(Gr_{e-\sum\alpha_{i}}(M^{\prime\prime})).

We count the dim of $R(M,e)$ by dividing it into two parts.

dimR(M,e)=dimb_{1(d_{1}-e_{1})}R(M,e)+dimR(M,e)/b_{1(d_{1}-e_{1})}R(M,e).

By lemma 7 and 8, since $b_{1(d_{1}-e_{1})}R(M,e)$ is acyclic,

dimb_{1(d_{1}-e_{1})}R(M,e)\leq dimR(M^{\prime\prime},e-\sum_{i\in I}\alpha_{i}).

So $dimR(M,e)=dimb_{1(d_{1}-e_{1})}R(M,e))+dimR(M,e)/b_{1(d_{1}-e_{1})}R(M,e)\\ \leq dimR(M^{\prime\prime},e-\sum_{i\in I}\alpha_{i})+dimR(M^{\prime},e)=\chi(Gr_{e}(M^{\prime})+\chi(Gr_{(e-\sum_{i\in I}\alpha_{i})}(M^{\prime\prime}))=\chi(Gr^{\Pi}_{e}(M))$ . ∎

Now we prove lemma 7 and 8.

Proof of lemma 7.

Recall $I(M)=\text{deg}\prod_{i\in I{+}}(t_{i})^{\gamma_{i}}\prod_{i\in I{-}}(s_{i})^{\gamma_{i}}\leq(\gamma,\nu)+D_{-\gamma}(M)$ We denote $v(M,\gamma,e)=(\gamma,\nu)+D_{w_{0}\gamma}(M)$ . The difference between $I(M)$ and $I(M)^{\prime}$ only occurs when $\gamma=-\varpi_{1}$ . In this case $v(M^{\prime},-\varpi_{1},e)=v(M,-\varpi_{1},e)-1$ . The degree of $s_{1}$ goes down by by 1, meaning we have one more vanishing condition which is $b_{1(d_{1}-e_{1})}=0$ . ∎

Proof of lemma 8.

In order to define a module structure on $b_{1(d_{1}-e_{1})}R(M,e)$ , we lift the element in $R(M^{\prime\prime},e-\sum_{i\in I}\alpha_{i})$ to $R(M,e)$ (since the former is a quotient of the latter) and let it act on $b_{1(d_{1}-e_{1})}R(M,e)$ by multiplication. We denote J to be the degree $v(M,\gamma,e)$ part of $<\prod_{i\in I{+}}(t_{i})^{\gamma_{i}}\prod_{i\in I{-}}(s_{i})^{\gamma_{i}},\gamma\in\Gamma>$ .

We need to check it is independent of the choice of the lift:

b_{1(d_{1}-e_{1})}J\subset I(M,e),\textup{ where }I(M,e)=J\oplus I(M^{\prime\prime},e-\sum_{i\in I}\alpha_{i}).

Denote the module corresponding to $i^{th}$ row P. We have $M=M^{\prime\prime}\oplus P$ . Then $v(M,\gamma,e)-v(M^{\prime\prime},\gamma,\sum_{i\in I}\alpha_{i})=v(P,\gamma,\sum_{i\in I}\alpha_{i})$ .
We claim that $v(P,\gamma,\sum_{i\in I}\alpha_{i})=0\ or\ 1$ and is 0 when $\gamma_{1}=1$ .
This is a direct calculation.

When $\gamma_{1}=-1$ , $t_{1j}$ appears in each summand $\prod_{i\in I{+}}(t_{i})^{\gamma_{i}}\prod_{i\in I{-}}(s_{i})^{\gamma_{i}},\gamma\in\Gamma$ . Let one of the summand be $t_{1j}k$ .
$s_{1(d_{1}-e_{1})}t_{1j}k=-\sum_{p+q=d_{1}-e_{1}+j}s_{1p}t_{1q}k=-s_{1(d_{1}-e_{1}+j-q)}\sum_{q>j}t_{1q}k$ . We have $\sum_{q>j}t_{1q}k$ is in $I(M,e)$ since this is of degree larger than $v(M,\gamma,e)$ .
When $\gamma_{1}=0$ ,
we claim that when $v(P,\gamma,\sum_{i\in I}\alpha_{i})$ is 1 , we have $\gamma-\varpi_{1}\in\Gamma$ .
Then we want to show $s_{1(d_{1}-e_{1})}\prod_{i\in I{+}}(t_{i})^{\gamma_{i}}\prod_{i\in I{-}}(s_{i})^{\gamma_{i}}$ is in $I(M,e)$ . Let k is a summand of degree $v(M,\gamma,e)$ part of $I(M,e)$ . We want to show $s_{1(d_{1}-e_{1})}k$ is of degree $v(M,\gamma-\varpi_{1},e)+1$ . So we need $v(M,\gamma,e)+d_{1}-e_{1}\geq v(M,\gamma-\varpi_{1},e)+1$ . By the dimension description, the image of $\Phi_{\gamma-\varpi_{1}}$ is at least 1 dimensional bigger than the image of $\Phi_{\gamma}$ since $\phi_{1m}(e_{i1})=e_{im}$ is in the image but for $\gamma-\varpi_{1}$ (since $\phi_{1m}$ is not a summand of $\phi$ ) the projection of $img(\phi)$ on $V_{m}$ is zero, where m is the smallest number in $\Gamma_{+}$ . ∎

Theorem 6.

$\Psi$ is an isomorphism when M is a kQ-module.

Proof.

by lemma 4, $\Psi$ is surjective so dim $k[a_{ij},b_{ik}]/I(M,e))\geq\chi(Gr^{\Pi}_{e}(M))$ and by lemma 5 this is an equality so the theorem follows. ∎

6 A consequence of this conjecture

In section 3, we defined $\mathcal{G}(G,Y)$ as moduli of maps of between pairs form $(d,d^{*})$ to $(G/Y,pt)$ . This is actually a local version of (fiber at a closed point c) the global loop Grassmannian with a condition Y to a curve $C$ , $\mathcal{G}^{C}(G,Y)$ . To a curve C, define $\mathcal{G}^{C}(G,Y)$ over the ran space $\mathcal{R}_{C}$ with the fiber at $E\in\mathcal{R}_{C}$ :

\mathcal{G}^{C}(G,Y)_{E}=^{def}map[(C,C-E),(G/Y,pt)].

Denote the map from $\mathcal{G}^{C}(G,Y)$ to $\mathcal{R}_{C}$ remembering the singularities by $\pi$ .

One can ask if $\pi$ is (ind) flat for any $G$ and $(Y,pt)$ . The case we are concerned is when $G^{\prime}=G\times\prod_{w}T_{w}$ and $Y=\prod_{w}(G/N^{w})^{aff}.$ Let $c\in C,\underline{\lambda_{w}},\underline{\mu_{w}}\in X_{*}(T)^{W}.$ In particular, we restrict $\mathcal{G}^{C}(G^{\prime},Y)$ to $C\times c$ and denote the image under projection from $\mathcal{G}(G^{\prime}$ to $\mathcal{G}(G)$ by $X$ . We have $X$ is a closed subscheme of $Gr_{G,X\times c}$ . Explicitly, an $R$ -point of $Gr_{G,X\times c}$ consists of the following data

•

$x:specR\xrightarrow[]{}C$ . Let $\Gamma_{x}$ be the graph of x. Let $\Gamma_{c}$ be the graph of the constant map taking value c.
•

$\beta$ a G-bundle on spec $R\times C$ .
•

A trivialization $\eta:\beta_{0}\xrightarrow[]{\eta}\beta$ defined on $specR\times C-(\Gamma_{x}\bigcup\Gamma_{c})$ .

An $R$ -point of $X$ over $C\times c$ consists of an $R$ -point of $Gr_{G,X\times c}$ subject to the condition: For every $i\in I$ , the composition

\eta_{i}:\beta_{0}\times^{G}V(\varpi_{i})\xrightarrow[]{}\beta\times^{G}V(\varpi_{i})\xrightarrow[]{}\beta\times^{G}V(\varpi_{i})\otimes\mathcal{O}(\langle\gamma,\lambda_{w}\rangle\cdot\Gamma_{x}+\langle\gamma,\mu_{w}\rangle\cdot\Gamma_{c}).

is regular on all of $specR\times C$ .

We can show the fiber over a closed point other than $c$ is $\bigcap\overline{S^{w}_{\lambda_{w}}}\times\bigcap\overline{S^{w}_{\mu_{w}}}$ and the fiber over $c$ is $\bigcap\overline{S^{w}_{\lambda_{w}+\mu_{w}}}$ .

Corollary 2 (Given the conjecture).

The T-fixed point subscheme of this family is flat.

Proof.

$dim\mathcal{O}((\overline{\bigcap S^{w}_{\lambda_{w}+\mu_{w}}}^{T})_{\nu})=dimH^{*}(Gr^{\Pi}_{e}(M))=\sum_{e_{1}+e_{2}=e}dimH^{*}(Gr^{\Pi}_{e_{1}}(M_{1})dimH^{*}(Gr^{\Pi}_{e_{2}}(M_{2})=\sum_{\nu_{1}+\nu_{2}=\nu}\mathcal{O}(\overline{\bigcap S^{w}_{\lambda_{w}}}^{T}_{\nu_{1}})\cdot\mathcal{O}(\overline{\bigcap S^{w}_{\mu_{w}}}^{T}_{\nu_{2}})=dim\mathcal{O}(\bigsqcup_{\nu_{1}+\nu_{2}=\nu}\overline{\bigcap S^{w}_{\lambda_{w}}}^{T}_{\nu_{1}}\times\overline{\bigcap S^{w}_{\mu_{w}}}^{T}_{\nu_{2}}=dim(\overline{\bigcup S^{w}_{\lambda_{w}}}^{T}\times\overline{\bigcap S^{w}_{\lambda_{w}}}^{T})_{\nu}.$ ⁸⁸8By lemma 6, and given the conjecture, Euler character is the same as total cohomology. ∎

Conjecture 2.

T-fixed subschemes flatness imply flatness.

We take $\lambda_{w}=-w_{0}\lambda+w\lambda$ , then $\overline{\bigcap S^{w}_{\lambda_{w}}}=\overline{Y^{\lambda}}$ . In this case the conjecture is proved to be true. This flatness is mentioned in [KMW16] remark 4.3 and will reduce the proof of reduceness of $\overline{Y^{\lambda}}$ to the case when $\lambda$ is $\varpi_{i}$ for each $i\in I$ .

7 Appendix

For an expression $s_{i_{m}}\cdots s_{i_{1}}$ of an element w in W, we say it is j-admissible if $\langle\alpha_{i_{a}},s_{i_{a-1}}\cdots s_{i_{1}}\varpi_{j}\geq 0$ for any $a\leq m$ .

Lemma 9.

For any element $w\in W$ , any reduced expression of $w$ is j-admissible.(since we will fix an j, we will omit j and just say admissible).

Proof.

Since we are in the ADE case,

	$\displaystyle s_{i}\varpi_{i}$	$\displaystyle=-\varpi_{i}+\sum_{h\text{ is adjacent to i }}\varpi_{h}.$		(6)
	$\displaystyle s_{i}\varpi_{h}$	$\displaystyle=\varpi_{h},\textup{ for $h\neq i$.}$		(7)

We use induction on the length of $w$ . Suppose lemma holds when $l(w)\leq m$ . Take a reduced expression of $w\in W$ with length $m+1$ : $w=s_{i_{m+1}}\cdots s_{i_{1}}$ . Suppose this expression is not admissible, we have $\langle\alpha_{i_{m+1}},s_{i_{m}}\cdots s_{i_{1}}\varpi_{j}\rangle\leq 0$ . Since $\langle\alpha_{i_{m+1}},\varpi_{j}\rangle\geq 0$ , and by (6),(7)

\langle\alpha_{i_{m+1}},s_{t}\gamma\rangle\geq\langle\alpha_{i_{m+1}},\gamma\rangle.

unless $t=i_{m+1}$ , there must exists $k$ such that $i_{k}=i_{m+1}$ .Let k be the biggest number such that $i_{k}=i_{m+1}$ .
In the case there is no element in the set $\{i_{m},\cdots,i_{k+1}\}$ is adjacent to $i_{m+1}$ in the Coxeter diagram, $s_{i_{m+1}}$ commutes with $s_{i_{m}}\cdots s_{i_{k+1}}$ . Therefore $s_{i_{m+1}}s_{i_{m}}\cdots s_{i_{k+1}}s_{i_{m+1}}=s_{i_{m+1}}s_{i_{m+1}}s_{i_{m}}\cdots s_{i_{k+1}}=s_{i_{m}}\cdots s_{i_{k+1}}$ so the $w=s_{i_{m+1}}\cdots s_{i_{1}}$ is not reduced, contradiction.
In the case where for some $t$ , $i_{t}$ is adjacent to $i_{m+1}$ , we will show we can reduce to the case we have only one such t. Suppose we have at least two elements $i_{t_{1}},i_{t_{2}}\cdots,i_{t_{h}}$ such that they are all adjacent to $i_{m+1}$ . Since $\langle\alpha_{i_{m+1}},s_{i_{m+1}}\cdots s_{i_{k}}\cdots s_{i_{1}}\varpi_{j}\rangle\leq 0$ and $h>1$ , by (6), (7), we must have some $i_{u_{1}},i_{u_{2}}$ such that they are adjacent to $i_{m+1}$ . Since one point at most has 3 adjacent points we must have some $i_{u_{x}}=i_{t_{y}}.$ Let $p=i_{u_{x}}=i_{t_{y}}$ . Using $s_{p}s_{i_{m+1}}s_{p}=s_{i_{m+1}}s_{p}s_{i_{m+1}}$ we can move $s_{i_{m+1}}$ in front of $s_{t_{y}}$ so the number h is reduced by 1. We could do this procedure until h=1. In this case we can rewrite the sequence before $s_{i_{k-1}}$ using the braid relation between $i_{m+1}$ and $i_{t}$ :

s_{i_{m}+1}\cdots s_{i_{t}}\cdots s_{i_{k}}=s_{i_{m}+1}\cdots s_{i_{t}}\cdots s_{{i_{m}+1}}=\cdots s_{i_{m+1}}s_{i_{t}}s_{i_{m+1}}\cdots=\cdots s_{i_{t}}s_{i_{m+1}}s_{i_{t}}\cdots.

Set $\beta=s_{i_{k-1}}\cdots s_{i_{1}}\varpi_{j}$ . By induction hypothesis, $s_{i_{m+1}}s_{i_{t}}s_{i_{m+1}}\cdots$ and $s_{i_{t}}s_{i_{m+1}}s_{i_{t}}\cdots$ are admissible. So $\langle\alpha_{i_{t}},\beta\rangle\geq 0$ and $\langle\alpha_{i_{m+1}},\beta\rangle\geq 0$ . Again using (6) and (7) we have $\langle s_{i_{t}}s_{i_{m+1}}\beta,\alpha_{i_{m+1}}\rangle\geq 0$ . Then $\langle s_{i_{m+1}}\cdots s_{i_{1}}\varpi_{j},\alpha_{i_{m+1}}\rangle=\langle s_{i_{t}}s_{i_{m+1}}\beta,\alpha_{i_{m+1}}\rangle\geq 0$ , contradicts with $s_{i_{m+1}}\cdots s_{i_{1}}$ is not admissible.

∎

Proof of lemma 2.

Set $F_{0}=\{\varpi_{j}\}$ . Let $F_{m}$ be the set which contains all $w\varpi_{j}$ , where $l(w)\leq m$ . We use induction. Suppose lemma 2 holds for $\gamma\in F_{m}$ , we will prove lemma holds when $\gamma\in F_{m+1}.$ For any $\gamma=w\varpi_{j}\in F_{m+1}$ , by lemma 1, w has a reduced admissible expression: $w=s_{i_{m+1}}\cdots s_{i_{1}}$ . Denote $i_{m+1}$ by i and $\beta$ by $s_{i_{m}}\cdots s_{i_{1}}\varpi_{j}$ . So $\gamma=s_{i}\beta$ , $\beta\in F_{m}$ . Since $s_{i_{m+1}}\cdots s_{i_{1}}$ is admissible, $\langle\beta,\alpha_{i}\rangle\geq 0$ . Therefore $\langle\gamma,\alpha_{i}\rangle=\langle s_{i}\beta,\alpha_{i}\rangle=-\langle\beta,\alpha_{i}\rangle\leq 0$ and we can apply prop 4.1 in [BK12]. Then $D_{\gamma}(M)=D_{s_{i}(s_{i}\gamma)}(M)=D_{s_{i}\gamma}(\Sigma_{i}M)$ , where $\Sigma_{i}$ is the reflection functor defined in section 2.2 in [BK12].
Let $A=\{j\ |\ j\text{ is adjacent to i, }j\in I\}$ and $M_{A}=\oplus_{s\in A}M_{s}$ . The $i^{th}$ component of $\Sigma_{i}M$ is the kernel of the map $\xi$ (Still see section 2.2 in [BK12] for the definition of $\xi$ ) from $M_{A}$ to $M_{i}$ . Since $\beta\in F_{m}$ , by induction hypothesis, we can apply this lemma to the case where $\gamma$ is taken to be $\beta$ and the module $M$ is $\Sigma_{i}M$ . Recall we denote by $I_{\gamma}^{+}$ the subset of I containing all i such that $\langle\gamma,\check{\alpha_{i}}\rangle$ is positive and by $I_{\gamma}^{-}$ containing all i $\langle\gamma,\check{\alpha_{i}}\rangle$ is negative.

Denote $A_{+}=\{j\ |\ j\text{ is adjacent to i, }j\in I_{\gamma}^{+}\}$ and $A_{-}=A\setminus A_{+}.$ For a multiset $S$ , let $M_{S}=\oplus M_{s}^{m(s)}$ . Regarding $I_{\gamma}^{-}$ as a multiset by setting $m(i)=\gamma_{i}^{-}$ , we can rewrite $\oplus_{i\in I_{\gamma}^{-}}M_{i}^{\gamma^{-}}$ as $M_{I^{-}_{\gamma}}$ , similarly $\oplus_{i\in I_{\gamma}^{+}}M_{i}^{\gamma^{+}}$ as $M_{I^{+}_{\gamma}}$ .

Consider the case when $\langle\gamma,\alpha_{i}\rangle=-1$ . We have $I_{\beta}^{+}=I_{s_{i}\gamma}^{+}=(I_{\gamma}^{+}\setminus A_{+})\bigcup\{i\}$ and $I_{\beta}^{-}=I_{s_{i}\gamma}^{-}=(I_{\gamma}^{-}\setminus\{i\})\bigcup A_{-}$ as multisets. Therefore $D_{s_{i}\gamma}(\Sigma_{i}M)$ is the dimension of the kernel the natural map (which is $\phi_{\beta}$ ) from $M_{I^{+}_{\gamma}\setminus A_{+}}\oplus ker(M_{A}\xrightarrow[]{\xi}M_{i})$ to $M_{I^{-}_{\gamma}\setminus\{i\}}\oplus M_{A_{-}}$ . This is equal to the dimension of the kernel of the natural map from $M_{I^{+}_{\gamma}\setminus A_{+}}\oplus ker(M_{A_{+}}\xrightarrow[]{\xi}M_{i})$ to $M_{I^{+}_{\gamma}\setminus\{i\}}$ , which is just $ker(M^{+}_{I_{\gamma}}\xrightarrow[]{\phi_{\gamma}}M_{I^{+}_{\gamma}})$ . The case when $\langle\gamma,\alpha_{i}\rangle=-2$ is similar.

∎

References

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University of Massachusetts, Amherst, MA.
E-mail address: zdong@math.umass.edu

loop grassmaniann

loop grassmaniann

A relation between Mirković-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE

Abstract

1 Introduction

2 Statement of the conjecture

2.1 Notation

2.2 MV cycles and polytopes

Theorem 1 ([Kam05]).

Proposition 1 ([Kam07]).

2.3 Objects on the quiver side

Proposition 2 ([Lus90], [BK12]).

2.4 A conjectural relation between MV cycles and modules over the preprojective algebra

Theorem 2 ([BK12]).

Conjecture 1.

3 The T fixed point subscheme of the cycle

Theorem 3.

Proof.

3.1 Ring of functions on (⋂Sλww¯)νT(\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}

Corollary 1.

4 Construction of the map Ψ\Psi from functions to cohomology

4.1 Map Ψ\Psi

Theorem 4.

4.2 Two lemmas

Lemma 1.

Lemma 2.

4.3 Proof of theorem 4 from lemmas in §\mathsection 4.2

Proof of theorem 4.

4.4 Recollection of GKM theory

Theorem 5 (GKM, see [Tym05], [GKM97]).

4.5 Affine paving of G​reΠ​(M)Gr^{\Pi}_{e}(M) when MM is a representation of QQ of type A

Definition 1 ([Tym07] 2.2).

Proposition 3 ([Tym07], 2.3).

Lemma 3.

Sublemma 1.

Proof.

Proof of lemma 3.

4.6 Proof of lemma 1

Proof.

5 Proof of isomorphism when MM is a representation of QQ of type A

Lemma 4.

Proof.

Definition 2.

Definition 3.

Proposition 4.

Lemma 5.

Lemma 6 (see prop 3.6 in [CC04]).

Lemma 7.

Lemma 8.

Proof of lemma 5.

Proof of lemma 7.

Proof of lemma 8.

Theorem 6.

Proof.

6 A consequence of this conjecture

Corollary 2 (Given the conjecture).

Proof.

Conjecture 2.

7 Appendix

Lemma 9.

Proof.

Proof of lemma 2.

References

3.1 Ring of functions on $(\bigcap\overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$

4 Construction of the map $\Psi$ from functions to cohomology

4.1 Map $\Psi$

4.3 Proof of theorem 4 from lemmas in $\mathsection$ 4.2

4.5 Affine paving of $Gr^{\Pi}_{e}(M)$ when $M$ is a representation of $Q$ of type A

5 Proof of isomorphism when $M$ is a representation of $Q$ of type A