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Archive for July, 2009

Calculation when level=rank=2

July 27, 2009

I will use the formula on the top of page 35 from Nakajima’s “Quiver Varieties and Branching”:
$\mathbf{w}=\sum w_i\Lambda_i = \sum \Lambda_{\mu_p}$ ; $\mathbf{w}-\mathbf{v}=\sum w_i\Lambda_i - v_i\alpha_i =\overline{\sp{t}\lambda}+t\delta^Y$ .
Where the first “t” in the second equation indicates transposing (according to the process described on page 33) the generalized young diagram $\lambda$ and the second “t” is given by the formula (on page 34):
$t=\langle d^X,\bar\mu\rangle-\langle d,M(\mu)\rangle$ .
$d^Y, \delta^Y, d^X, \delta^X$ are suitable choices for “d” and “ $\delta$ ” in affine $\tilde{L\mathfrak{sl}_2}$ (which in this case is on “both sides” of the level-rank duality).
And $\langle d, M(\mu)\rangle$ is (I believe) the coefficient of M in the formula for d(M) found near the bottom of page 32.

My goal is to find the cycles corresponding to the first two triangles in my list of polytopes so I use:
$\mu=\delta^X+\Lambda_0+\Lambda_1$
$\overline\lambda=\overline{\sp{t}\lambda}=\Lambda_0+\Lambda_1$
$\mu_1=0, \mu_1=-1$
Some of the notation is very confusing, I understand. I’m sorry, I don’t know what to do with it: at the end of the day the dimension vectors we are concerned about are $\mathbf{w}=(1,1)$ and $\mathbf{v}=(1,1)$

$\mathbb{C}$	$\leftarrow$	$\mathbb{C}$
	$\rightarrow$
	$\rightarrow$
	$\leftarrow$
$\downarrow\uparrow$		$\downarrow\uparrow$
$\mathbb{C}$		$\mathbb{C}$

We want to replace the arrows in this diagram with maps (in order of appearance top to bottom left to right) $y,\bar y,x,\bar x,b_1,a_1,b_2,a_2$ in such a way that it satisfies three conditions which I abbreviate as the “ $\mu^{-1}(0)$ ” (this is not the same $\mu$ … sorry), “stability” and “limit” conditions.

The “ $\mu^{-1}(0)$ ” condition in our case (for and for other choices of $\mathbf w \mathbf v$ but still r=l=2) implies:
$a_1b_1+\bar x x=y\bar y$
$a_2b_2+\bar y y=x\bar x$

The “Stability” condition has two halves. First that every vector v in the top half has “Origins” i.e. $v=\sum f_k(\eta_k)$ where $\eta_k$ live in the bottom half and $f_k$ are chosen from appropriate paths in the quiver. And second that every non-zero v in the top half has “Futures” i.e. $g(v)\neq 0$ for some path g terminating in the bottom half.

The “Limit” condition requires that the action of t (not the same as either previous t… sorry) given on page 30 can be “controled” as $t\rightarrow\infty$ by action of $G_V$ described on page 5 (just before eq. 2.1 )

Something to notice right away because it will be useful in future calculations: taking the two halves of “Stability” together gives paths $g\circ f:W^j\rightarrow W$ on which $G_V$ has no control! The “Limit” condition requires that the image of such a composition must lie in $W^{<j}$ . (Superscript refers to the decomposition into 1-dimensional subspaces given by $\mu$ — use of the $\le$ is short hand for the corresponding filtration.)

In our case we have $b_2=0,a_1=0$ . For example if the image of $a_1$ is non-zero the "Futures" condition says it must escape but, according to the "Limit" condition, in doing so it can only afford to accumulate $m_1$ orders of t and every possible exiting accumulates at least $m_1+1$ . The argument for $b_2$ is quite similar replacing "image" with "kernel", "Futures" with "Origins" etc..

Furthermore we get:
$a_1\neq0\neq b_2$ ,
$\bar y=0\Rightarrow x\neq 0$ and
$\bar y\neq 0\Rightarrow y=0$ .
(Moding out by the $G_V$ action in this case we can assume $a_1=1=b_2:\mathbb{C}\rightarrow\mathbb{C}$

The two components, then, correspond to diagrams:

$\mathbb{C}$	$\bar y\rightarrow$	$\mathbb{C}$
$\mathbb{C}$	$x\dashrightarrow$	$\mathbb{C}$
$\downarrow$		$\uparrow$
$\mathbb{C}$		$\mathbb{C}$

and

$\mathbb{C}$	$\dashleftarrow y$	$\mathbb{C}$
$\mathbb{C}$	$x\rightarrow$	$\mathbb{C}$
$\downarrow$		$\uparrow$
$\mathbb{C}$		$\mathbb{C}$

Where the solid line indicates the map is non-zero and dashed line indicates the map may be zero.

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GRTEALA 1: Review of the situation

July 8, 2009

Update: Notes and video from the summer school is available here.

I should have been posting while I was at the conference. But anyway I’ll try to post as much as I can remember, before I forget it.

Geometric Satake gives a correspondence from representation theory to subvarieties of an affine Grassmanian. MV-cycles help give a better handle on them but are still rather abstract. MV-polytopes , introduced by Jared Anderson, are more “hands on” in terms of being able to do direct computations. But you need to know what they are first, and their original definition as moment map images of MV-cycles doesn’t really help. At least if you know they are a convex hill of the torus fixed-points appear in the *closure* of an MV-cycle then you’re done — but this still requires more-or-less calculating the MV-cycles and taking their closure.

Kamnitzer’s thesis provided a few ways to get direct handle on MV-polytopes avoiding MV-cycles entirely.

Implicit description via Plucker relations
Inductive description (for $\mathfrak{sl}_n$ later extended to types B and C by **FIXME**)
Reduction to dim-2: higher dimensional MV-polytopes are all polytopes whose 2-faces are MV-polytopes.
Construction from primitives: MV-polytopes are Minkowski sums of “primitives” and sums of primitives from the same “cluster” are MV-polytopes
In dimension 2, clusters can be described by networks of non-overlapping cords each parallel to a sides of the Weyl polytope

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A conjecture on the Uniqueness of MV-Polytopes

July 6, 2009

Fix a lattice L with $\langle , \rangle$ and a partial order $\le$ . Define
$Q^+=\{x\in L|x\ge 0\} = \text{``positive root cone''}$
$L^+=\{x\in L|\langle x,y\rangle \ge 0 \forall y\in Q^+\} = \text{``dominant weights''}$
(Alternatively we could fix $Q^+$ and define $x\ge y \iff x-y\in Q^+$ . Similarly we might fix L^+ without reference to any inner product.)

We say that a set of “characters” $\{\chi_\lambda:L\rightarrow\mathbb{N}\}_{\lambda\in L^+}$ and a set of subsets $\{M^\lambda_\mu \subset L\}_{\lambda,\mu\in L}$ satisfy the “tensor property” if:

$\chi_\lambda(\mu)=\#\{m\in M^\lambda_\mu | m\subset \text{supp} \chi_\lambda\}$
$\chi_{\lambda_1} * \chi_{\lambda_2}=\sum c^{\lambda_1,\lambda_2}_\mu \chi_\mu$ where
$c^{\lambda_1,\lambda_2}_\mu= \#\{ m \in M^{\lambda_1+\lambda_2}_\mu | m\subset ((\text{supp}\chi_{\lambda_1})+\lambda_2)\cap(\mu+\lambda_2-\text{supp}\chi_{\lambda_2}))\}$

(Here $*$ indicates convolution and $\mathbb{N}$ includes zero.)

Suppose we also require the following about $\{M^\lambda_\mu\}$

$M^\lambda_\mu\neq\o \iff \lambda\ge\mu$
$m\in M^\lambda_\mu\Rightarrow \mu\le x\le\lambda \forall x\in m \text{ and } \mu,\lambda\in m$
$m\in M^\lambda_\mu\Rightarrow m+a\in M^{\lambda+a}_{\mu+a} \forall a\in L \text{ and } (-m)\in M^{-\mu}_{-\lambda}$
(*) $\sigma a\in M^{\sigma\lambda}_{\sigma\mu}$ for any isotopy, $\sigma$ of $Q^+$ .

(I’m not sure if the last condition is worded correctly, but I’d like the collection to be invariant under permutations of, for example, minuscule weights.)

And, fixing a group W acting on L for which $L^+$ is a fundamental domain, suppose we require that:

$\chi_\lambda(\mu)=\chi_\lambda(w\mu) \forall w\in W$

Conjecture For $L,Q^+,L^+,W$ there is only one $\{M^\lambda_\mu\}_{\lambda,\mu\in L}$ (and corresponding $\{ \chi_\lambda\}_{\lambda\in L^+}$ ) satisfying the tensor property and all the above requirements.