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Documentation for "GKMVarieties.m2"
Authors: Christopher Eur, Ritvik Ramkumar
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beginDocumentation()
doc ///
Key
GKMVarieties
Headline
computations with GKM varieties and moment graphs
Description
Text
A GKM variety is a variety $X$, often assumed to be smooth and complete, with an
action of an algebraic torus $T$ satisfying the following conditions:
(i) $X$ is equivariantly formal with respect to the action of $T$,
(ii) $X$ has finitely many $T$-fixed points, and (iii) $X$ has finitely
many one-dimensional $T$-orbits. The data of the zero and one dimensional
$T$-orbits of $X$ define the moment graph of $X$, with which one can carry out
$T$-equivariant cohomology and $T$-equivariant $K$-theory computations via
the method of localization.
This package provides methods for these computations in Macaulay2.
Text
For mathematical background see:
@UL{
{"[BM01] T. Braden and R. MacPherson. From moment graphs to intersection cohomology. Math. Ann. 321 (2001), 533-551."},
{"[BGH02] E. Bolker, V. Guillemin, and T. Holm. How is a graph like a manifold? arXiv:math/0206103."},
{"[CDMS18] A. Cameron, R. Dinu, M. Michalek, and T. Seynnaeve. Flag matroids: algebra and geometry. arXiv:1811.00272."},
{"[DES20] R. Dinu, C. Eur, and T. Seynnaeve. K-theoretic Tutte polynomials of morphisms of matroids. arXiv:math/2004.00112."},
{"[FS12] A. Fink and S. Speyer. K-classes for matroids and equivariant localization. Duke Math. J. 161 (2012), no. 14, 2699-2723."},
{"[GKM98] M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131 (1998), no. 1, 25-83."},
{"[RK03] I. Rosu. Equivariant K-theory and equivariant cohomology. With an Appendix by I. Rosu and A. Knutson. Math. Z. 243 (2003), 423-448."},
{"[Tym05] J. Tymoczko. An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson. Contemp. Math. 388 (2005), 169-188."},
{"[VV03] G. Vezzosi and A. Vistoli. Higher algebraic K-theory for actions of diagonalizable groups. Invent. Math. 153 (2003), no. 1, 1–44."}
}@
Text
@SUBSECTION "Contributors"@
Text
The following people have contributed code, improved existing code, or enhanced the documentation:
@HREF("https://www.mis.mpg.de/combag/members/tim-seynnaeve.html","Tim Seynnaeve")@.
SeeAlso
"Example: generalized flag varieties"
"Example: smooth toric varieties"
///
doc ///
Key
"Example: generalized flag varieties"
Description
Text
Let $G$ be a reductive complex Lie group and $P$ a
parabolic subgroup containing a maximal torus $T$. The generalized flag variety $G/P$ is a GKM variety
with the action of $T$. This package allows users to create a generalized flag variety for classical Lie types
($A$, $B$, $C$, and $D$) as a @TO "GKMVariety"@ with conventions explicitly laid out as follows.
Text
For type $A_{n-1}$, the group $G$ is $GL_{n}$, and the torus $T$ is $diag(t_1, \ldots, t_n)$, the group of invertible diagonal matrices.
Text
For type $B_n$, the group $G$ is $SO_{2n+1}$, where we set the standard symmetric bilinear form on $\mathbb C^{2n+1}$
to be is given by the matrix
$$\begin{pmatrix} 0 & I_n & 0 \\ I_n & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
and the torus $T$ is $diag(t_1, \ldots,t_n, t_1^{-1}, \ldots, t_n^{-1}, 1)$.
Text
For type $C_n$, the group $G$ is $Sp_{2n}$, where we set the standard alternating bilinear form on
$\mathbb C^{2n}$ to be given by the matrix
$$\begin{pmatrix} 0 & -I_n \\ I_n & 0 \end{pmatrix}$$
and the torus $T$ is $diag(t_1, \ldots,t_n, t_1^{-1}, \ldots, t_n^{-1})$.
Text
For type $D_n$, the group $G$ is $SO_{2n}$, where we set the standard symmetric bilinear form on
$\mathbb C^{2n}$ to be given by the matrix
$$\begin{pmatrix} 0 & I_n \\ I_n & 0 \end{pmatrix}$$
and the torus $T$ is $diag(t_1, \ldots,t_n, t_1^{-1} \ldots, t_n^{-1})$.
Text
In all the cases, the standard action of $(\mathbb C^*)^m$ on $\mathbb C^m$ is defined by $(t_1, \ldots, t_m) \cdot (x_1, \ldots, x_m) = (t_1^{-1}x_1, \ldots, t_m^{-1}x_m)$.
Text
Let $\{w_1, \ldots, w_n\}$ be a set of fundamental weights, which for classical Lie types are explicitly set to be
as follows:
($A_{n-1}$): $\{w_1, \ldots, w_n\}= \{e_1, e_1+e_2, \ldots , e_1+e_2+\cdots+e_{n-1}\}$
($B_n$): $\{w_1, \ldots, w_n\}= \{e_1, e_1+e_2, \ldots , e_1+\cdots+e_{n-1}, (1/2)(e_1+\cdots e_n)\}$
($C_n$): $\{w_1, \ldots, w_n\}= \{e_1, e_1+e_2, \ldots , e_1+\cdots+e_{n-1}, e_1 + \cdots +e_n\}$
($D_n$): $\{w_1, \ldots, w_n\}= \{e_1, e_1+e_2, \ldots , e_1+\cdots+e_{n-2}, (1/2)(e_1+\cdots+e_{n-2} +e_{n-1}- e_{n}), (1/2)(e_1+\cdots+e_{n-2}+e_{n-1}+e_n)\}$
Text
For a sequence $(a_1, \ldots, a_n)\in \mathbb N^n$ of nonnegative integers,
let $I = \{i \mid a_i \neq 0\}$ and $P_I$ the corresponding parabolic subgroup of $G$.
Then the generalized flag variety $G/P_I$ is embedded in the irreducible representation of $G$
with the highest weight $a_1w_1 + \cdots a_nw_n$.
These generalized flag varieties can be created as a @TO "GKMVariety"@ using the method
@TO generalizedFlagVariety@. For instance, the Grassmannian $Gr(2,4)$ of
2-dimensional subspaces in $\mathbb C^4$, embedded in $\mathbb P^5$ by the usual Plücker embedding,
can be created as follows.
Example
Gr24 = generalizedFlagVariety("A",3,{2})
peek Gr24
Text
The @TO MomentGraph@ of $Gr(2,4)$ is the 1-skeleton of the hypersimplex $\Delta(2,4)$, a.k.a. the octahedron.
Example
G = momentGraph Gr24
underlyingGraph G
Text
The line bundle $O(1)$ on $Gr(2,4)$, corresponding to its Plücker embedding, can be accessed by @TO (ampleKClass, GKMVariety)@.
The method @TO (euler, KClass)@ computes its Lefschetz trace (a.k.a. equivariant Euler characteristic),
which in this case is the Laurent polynomial in the character ring of the torus $T$
whose terms correspond to be weights of the second exterior power of the standard representation of $GL_4$.
Example
O1 = ampleKClass Gr24 --the O(1) bundle on Gr24 via its Plücker embedding
euler O1
Text
If $Gr(2,4)$ is embedded differently, say by the line bundle $O(2)$ instead, the Lefschetz trace changes
accordingly, and its coefficients record the multiplicities of the associated weight spaces in the second
symmetric power of the second exterior power of the standard representation of $GL_4$.
Example
euler (O1^2)
Text
The Schubert decomposition of $Gr(2,4)$, and more generally the Bruhat decomposition of $G/P$, can be accessed
by the method @TO (bruhatOrder, GKMVariety)@, which outputs the poset of the Bruhat order. Moreover, the Schubert
varieties can be created via the method @TO generalizedSchubertVariety@.
Example
P1 = bruhatOrder Gr24
Sch = generalizedSchubertVariety(Gr24,{set{1,2}})
P2 = bruhatOrder Sch
--{P1,P2}/displayPoset --to view the posets in pdf
Text
The "forgetful" map from the complete flag variety $Fl(4)$ to $Gr(2,4)$,
given by forgetting the subpsaces in the complete flag except for the 2-dimensional one, can be created as
a @TO EquivariantMap@ by the method @TO flagMap@.
Example
Fl4 = generalizedFlagVariety("A",3,{1,2,3},Gr24.characterRing) --Fl(4) with the torus having the same character ring as Gr24
f = flagMap(Fl4,Gr24)
Fl4 === f.source and Gr24 === f.target
Text
As $Fl(4)$ is a $BiProj$ of vector bundles on $Gr(2,4)$, the (derived) pushforward of the structure sheaf
of $Fl(4)$ is the structure sheaf of $Gr(2,4)$ since the higher direct images vanish under the forgetful map.
Example
(trivialKClass Gr24) === (pushforward f)(trivialKClass Fl4)
Text
For type $C$, the following example features the isotropic Grassmannian $SpGr(2,6)$ consisting of
2-dimensional subspaces in $\mathbb C^6$ that are isotropic with respect to the standard alternating form.
The vertices of its moment graph can be considered as the vertices of the cuboctahedron.
Example
SpGr26 = generalizedFlagVariety("C",3,{2})
peek SpGr26
momentGraph SpGr26
Text
The second fundamental representation of $Sp_{6}$ is 14-dimensional with 12 extremal weights.
Example
euler ampleKClass SpGr26
Text
For type $B$, the following example features the isotropic Grassmannian $SOGr(2,5)$ consisting of
3-dimensional subspaces in $\mathbb C^5$ that are isotropic with respect to the standard symmetric form.
Its moment graph is the a complete graph on 4 vertices.
Note that Spin groups and their representations are not implemented, so for the type $B_n$ the coefficient
$a_n$ need be a multiple of 2.
Example
SOGr25 = generalizedFlagVariety("B",2,{2,2}) --inputing {2} instead of {2,2} results in error: spin groups not implemented yet
peek SOGr25
euler ampleKClass SOGr25
Text
For type $D$, the following example features the isotropic Grassmannian $SOGr(3,8)$ consisting of
3-dimensional subspaces in $\mathbb C^8$ that are isotropic with respect to the standard symmetric form.
Example
SOGr38 = generalizedFlagVariety("D",4,{3,4})
SOGr38.points
Text
Similarly as in type $B$, Spin groups are not implemented, so the two connected components of
$SOGr(4,8)$ need be separatedly created in the following way.
Example
SOGr48odd = generalizedFlagVariety("D",4,{3,3})
SOGr48odd.points
SOGr48even = generalizedFlagVariety("D",4,{4,4})
SOGr48even.points
Caveat
Does not check for low-dimensional isogenies. For instance, always use type $D_n$ with $n\geq 4$ to be safe.
SeeAlso
generalizedFlagVariety
flagMap
GKMVariety
makeGKMVariety
///
doc ///
Key
"Example: smooth toric varieties"
Description
Text
A toric variety is an integral variety containing an open dense algebraic torus. If the
toric variety is smooth (or simplicial) it is naturally a GKM variety:
Let $X$ be a smooth toric variety
and $U$ be an affine chart whose associated character lattice is generated by elements of
weights $a_1,\dots, a_m$. Then $(\mathbb C^*)^n$ acts on $U$ by
$t \cdot (x_1,\dots, x_n) = (t^{-a_1}x_1,\dots, t^{-a_n}x_n)$.
We caution that this package uses the outer normals instead of inner normals.
Text
The method @TO "normalToricVariety"@ from the package @TO "NormalToricVarieties"@ allows the
user to construct smooth toric varieties. To convert it to a GKM variety we use the method
@TO "makeGKMVariety"@. Here is an example with $X = Bl_p\mathbb P^2$, the blow-up of $\mathbb P^2$ at a point,
which is also the first Hirzebruch surface.
Example
FF1 = hirzebruchSurface 1;
X = makeGKMVariety FF1;
peek FF1
peek X
Text
If a GKM variety $X$ was originally constructed from @TO "normalToricVariety"@ we can convert it
back to a toric variety.
Example
Y = normalToricVariety(X); -- X defined in the previous example above
Y === FF1
Text
Continuing this example, the following shows how to convert a torus-invariant divisor constructed using
@TO ToricDivisor@ to a @TO KClass@.
Example
antiK = - toricDivisor(FF1) -- the anti-canonical class on FF1
TantiK = makeKClass(X,antiK)
isWellDefined TantiK
Text
Since the toric variety $X = Bl_p\mathbb P^2$ is Gorenstein Fano,
with its anticanonical embedding in $\mathbb P^8$, the equivariant Euler characteristic of
the anticanonical divisor is the sum of the characters of the sections of the associated line bundle.
Example
euler TantiK
Text
We caution the following difference in convention:
Projective $n$-space $\mathbb P^n$ as a @TO NormalToricVariety@ constructed using @TO "toricProjectiveSpace"@
is acted upon by an $n$-dimensional torus. However, as a @TO GKMVariety@
constructed using @TO "projectiveSpace"@, it is acted upon by an $(n+1)$-dimensional torus.
Example
X = makeGKMVariety toricProjectiveSpace 2; -- the torus is C^2
Y = projectiveSpace 2; -- the torus is C^3
peek X
peek Y
SeeAlso
(makeGKMVariety, NormalToricVariety)
projectiveSpace
normalToricVariety
(makeKClass, GKMVariety, ToricDivisor)
///
doc ///
Key
(normalToricVariety, GKMVariety)
Headline
converts a GKM variety back into a toric variety
Usage
Y = normalToricVariety X
Inputs
X:GKMVariety
Outputs
Y:NormalToricVariety
Description
Text
If $X$ is a GKM variety that was originally constructed using @TO "normalToricVariety"@, then
this method reverts $X$ to a @TO "NormalToricVariety"@.
Example
X = toricProjectiveSpace 2;
Y = makeGKMVariety X
assert(normalToricVariety Y === X)
SeeAlso
makeGKMVariety
normalToricVariety
///
doc ///
Key
GKMVariety
Headline
the class of all GKM varieties
Description
Text
A @TO GKMVariety@ $X$ is a @TO MutableHashTable@ representing a GKM variety $X$ with an action of a torus $T$.
Its keys include:
@UL{
{TT "points", ", whose value is a list representing the torus-fixed points of ", TEX "$X$"},
{TT "characterRing", ", whose value is a ring representing the character ring of ", TEX "$T$"},
{TT "momentGraph", ", whose value is the ", TO "MomentGraph", " of ", TEX "$X$"},
{TT "charts", ", whose value is a ", TO "HashTable", " representing the (negatives of) characters of the torus action
on each torus-invariant affine chart around a torus-fixed point. ", " The keys of ", TT "X.charts", " are ", TT "X.points",
" and the values are lists consisting of lists of integers."}
}@
Text
Every @TO GKMVariety@ created by methods in this package has at least the two keys @TT "points"@ and @TT "characterRing"@.
The following example is the projective space $\mathbb P^2$ as a @TO GKMVariety@.
Example
PP2 = projectiveSpace 2
peek PP2
SeeAlso
makeGKMVariety
"Example: generalized flag varieties"
"Example: smooth toric varieties"
///
doc ///
Key
makeGKMVariety
(makeGKMVariety, List, Ring)
(makeGKMVariety, List, List, Ring)
(makeGKMVariety, MomentGraph)
(makeGKMVariety, MomentGraph, Ring)
(makeGKMVariety, NormalToricVariety)
(makeGKMVariety, NormalToricVariety,Ring)
(makeGKMVariety, KClass)
Headline
constructs a GKM variety
Usage
X = makeGKMVariety(L,R)
X = makeGKMVariety(L,M,R)
X = makeGKMVariety(G)
X = makeGKMVariety(G,R)
X = makeGKMVariety(R)
X = makeGKMVariety(Y,R)
X = makeGKMVariety(C)
Inputs
L:List
of torus-fixed points of $X$
M:List
of lists; the i-th list consists of the (negatives of) characters of the
action of the torus on a torus-invariant affine chart around the torus-fixed point
corresponding to L_i
G:MomentGraph
representing the one dimensional torus-orbits of $X$
R:Ring
representing the character ring of the torus acting on $X$
Y:NormalToricVariety
C:KClass
Outputs
X:GKMVariety
Description
Text
The minimum data needed to create a @TO "GKMVariety"@ are the set of torus-fixed points
and the character ring. Here is an example with projective space
Example
L = {0,1,2,3};
R = makeCharacterRing 4
X = makeGKMVariety(L,R)
Text
If necessary, we can add the (negatives of) characters of the action of the torus on each
torus-invariant chart of $X$. Note that the i-th entry of the list below corresponds to
the i-th entry of L.
Example
M = {{{-1, 1, 0, 0}, {-1, 0, 1, 0}, {-1, 0, 0, 1}},
{{1, -1, 0, 0}, {0, -1, 1, 0}, {0, -1, 0, 1}},
{{1, 0, -1, 0}, {0, 1, -1, 0}, {0, 0, -1, 1}},
{{1, 0, 0, -1}, {0, 1, 0, -1}, {0, 0, 1, -1}}};
Y = makeGKMVariety(L,M,R);
peek Y
Text
To produce one of the generalized flag varieties we use the method @TO generalizedFlagVariety@
Here is an example of the Lagrangian Grassmannian $SpGr(2,4)$ consisting of 2-dimensional subspaces
in $\mathbb C^4$ that are isotropic with respect to the standard alternating form.
Example
SpGr24 = generalizedFlagVariety("C",2,{2})
peek SpGr24
Text
Here is the complete flag variety of $Sp_4$.
Example
SpFl4 = generalizedFlagVariety("C",2,{1,2})
peek SpFl4
Text
The following example produces the Orthogonal Grassmannian $SOGr(2,5)$ from its
moment graph.
Example
V = {{set {0, 1}}, {set {0, "1*"}}, {set {"0*", 1}}, {set {"0*", "1*"}}};
edgs = {{{set {"0*", 1}}, {set {"0*", "1*"}}},
{{set {0, "1*"}}, {set {"0*", "1*"}}},
{{set {0, "1*"}}, {set {"0*", 1}}},
{{set {0, "1*"}}, {set {0, 1}}},
{{set {0, 1}}, {set {"0*", "1*"}}},
{{set {0, 1}}, {set {"0*", 1}}}};
wghts = {{0,-1},{-1,0},{-1,1},{0,1},{-1,-1},{-1,0}}
E = hashTable(apply(edgs, v -> (v,wghts)));
t = symbol t; H = QQ[t_0, t_1]
G = momentGraph(V,E,H);
Z = makeGKMVariety(G);
peek Z
Caveat
This function does not check if X is a valid GKM variety.
SeeAlso
(symbol **, GKMVariety, GKMVariety)
projectiveSpace
generalizedFlagVariety
(map, GKMVariety, GKMVariety, List)
///
doc ///
Key
(symbol **, GKMVariety, GKMVariety)
Headline
product of GKM varieties
Usage
X ** Y
Inputs
X:GKMVariety
Y:GKMVariety
Outputs
X:GKMVariety
product of X and Y
Description
Text
Given two GKM varieties $X$ and $Y$ with an action of a common torus $T$, the
product is $X \times Y$ with the structure of a GKM variety given by the
diagonal action of $T$. This method constructs $X \times Y$ as a @TO "GKMVariety"@.
To speed up computation, this method does not automatically cache
the moment graph of $X \times Y$. The user can cache this using the method
@TO"MomentGraph ** MomentGraph"@.
Text
The following example exhibits the product of $\mathbb P^1$ with
the Lagrangian Grassmannian SpGr(2,4).
Example
R = makeCharacterRing 2;
X = projectiveSpace(1,R);
Y = generalizedFlagVariety("C",2,{2},R);
Z = X ** Y;
peek Z
Text
We can cache the moment graph of $Z$ as follows:
Example
G = momentGraph X;
H = momentGraph Y;
momentGraph(Z, G** H);
peek Z
SeeAlso
makeGKMVariety
///
doc ///
Key
KClass
Headline
the class of all equivariant K-classes
Description
Text
For $X$ a GKM variety with an action of a torus $T$ whose character ring is $R$,
a $T$-equivariant $K$-class $C \in K_T^0(X)$ of is encoded by its image in $K_T^0(X^T) = \prod_{x\in X^T} R$,
under the injective restriction map $K_T^0(X) \to K_T^0(X^T)$.
See [Corollary 5.12; VV03] or [Corollary A.5; RK03] for details.
Text
A @TO "KClass"@ C is a @TO "HashTable"@
consisting of two keys:
@UL{
{TT "variety", ", whose value is a ", TO "GKMVariety", " of which C is a K-class of"},
{TT "KPolynomials", ", whose value is a ", TO "HashTable", "; its keys are ", TT "X.points", " and the values are
Laurent polynomials in the character ring representing the values of the K-class under the restriction map."}
}@
SeeAlso
makeKClass
(isWellDefined, KClass)
pushforward
pullback
(euler, KClass)
///
doc ///
Key
makeKClass
(makeKClass, GKMVariety, List)
Headline
constructs an equivariant K-class
Usage
C = makeKClass(X,L)
Inputs
X:GKMVariety
L:List
of Laurent polynomials corresponding to each torus-fixed point
D:ToricDivisor
Outputs
C:KClass
Description
Text
This method creates a @TO KClass@ given a @TO GKMVariety@ @TT "X"@ and a list @TT "L"@ of Laurent polynomials in its
character ring. The order of Laurent polynomials in the list must correspond to the order of the list
of torus-fixed points @TT "X.points"@.
Text
The following example is the class of $O(1)$ on the projective space $\mathbb P^3$.
Example
PP3 = projectiveSpace 3;
R = PP3.characterRing;
L = gens R
C = makeKClass(PP3,L) --the class of O(1) on PP3
C === ampleKClass PP3
isWellDefined C
Caveat
This function does not check if X defines a GKM variety - see
@TO2{(isWellDefined, KClass), "isWellDefined"}@.
SeeAlso
(isWellDefined, KClass)
(symbol *, KClass, KClass)
(symbol +, KClass, KClass)
pullback
pushforward
///
doc ///
Key
(makeKClass, GKMVariety, ToricDivisor)
Headline
create the KClass from a ToricDivisor
Usage
C = makeKClass(X,D)
Inputs
X:GKMVariety
created from a @TO NormalToricVariety@
D:ToricDivisor
Outputs
C:KClass
Description
Text
If a GKM variety $X$ also admits a structure of a @TO NormalToricVariety@,
then the following example shows how to obtain the @TO KClass@ of any
@TO ToricDivisor@ on $X$.
Example
X = toricProjectiveSpace 3;
D = toricDivisor({1,0,0,0},X) -- the class of O(1) on P^3
Y = makeGKMVariety X; -- The torus is C^3 not C^4
C = makeKClass(Y,D)
assert(isWellDefined C)
peek C
Caveat
Toric vector bundles are yet to be imported.
SeeAlso
makeKClass
KClass
(normalToricVariety, GKMVariety)
(makeGKMVariety, NormalToricVariety)
///
doc ///
Key
(isWellDefined, KClass)
Headline
whether the input is a well-defined equivariant K-class
Usage
isWellDefined C
Inputs
C:KClass
Outputs
:Boolean
whether or not a list of Laurent polynomials satisfies edge compatibility condition
--prints out the edges of the moment graph for which @TT "C"@ fails the compatibility condition
Description
Text
If $\{f_x \mid x\in X^T\}$ is a collection of Laurent polynomials in the
character ring $\mathbb Z[T_0, \ldots, T_n]$ of the torus $T$ acting on a @TO GKMVariety@ $X$, one per each torus-fixed point, representing an element $C$ of $K_T^0(X^T)$,
then $C$ is in the image of $K_T^0(X)$ under the injective restriction map $K_T^0(X)\to K_T^0(X^T)$ if and only if
it satisfies the following "edge compatibility condition":
For each one-dimensional $T$-orbit-closure in $X$ with boundary points $x$ and $x'$, one has
$$f_x \equiv f_{x'} \ \mod \ 1 - T^{\lambda(x,x')}$$
where $\lambda(x,x')$ is the character of the action of $T$ on the
one-dimensional orbit.
See [Corollary 5.12; VV03] or [Corollary A.5; RK03] for details.
Example
PP3 = projectiveSpace 3
isWellDefined ampleKClass PP3 --the O(1) class on PP3 is well-defined
badC = makeKClass(PP3, reverse gens PP3.characterRing) --reverse the order of Laurent polynomials defining the O(1) class
isWellDefined badC --no longer well-defined
Caveat
A @TO MomentGraph@ must be defined on the @TO GKMVariety@ on which the @TO KClass@ is a $K$-class of.
SeeAlso
KClass
makeKClass
///
doc ///
Key
(symbol *, KClass, KClass)
Headline
computes the product of two equivariant K-classes
Usage
C1 * C2
Inputs
C1:KClass
C2:KClass
Outputs
:KClass
the product of C1 and C2
Description
Text
This method computes the product of two equivariant $K$-classes.
Example
Gr24 = generalizedFlagVariety("A",3,{2}); --the Grassmannian of projective lines in projective 3-space
O1 = ampleKClass Gr24 -- the O(1) bundle on Gr24 as an equivariant K-class
O2 = O1 * O1
peek O2
SeeAlso
makeKClass
(symbol ^, KClass, ZZ)
(symbol +, KClass, KClass)
///
doc ///
Key
(symbol ^, KClass, ZZ)
Headline
computes powers of an equivariant K-classes
Usage
C^n
Inputs
C:KClass
n:ZZ
Outputs
:KClass
the n-th power of C
Description
Text
This method computes the $n$-th power of an equivariant $K$-class $C$.
Example
Gr24 = generalizedFlagVariety("A",3,{2}); --the Grassmannian of projective lines in projective 3-space
O1 = ampleKClass Gr24 -- the O(1) bundle on Gr24 as an equivariant K-class
O2 = O1^2
peek O2
Oneg1 = O1^(-1)
peek Oneg1
Caveat
$n$ is allowed to be negative only when $C$ is a line bundle, or a direct sum of copies of a line bundle.
SeeAlso
makeKClass
(symbol *, KClass, KClass)
(symbol +, KClass, KClass)
///
doc ///
Key
(symbol +, KClass, KClass)
Headline
computes the sum of two equivariant K-classes
Usage
C1 + C2
Inputs
C1:KClass
C2:KClass
Outputs
:KClass
the sum of C1 and C2
Description
Text
This method computes the sum of two equivariant $K$-classes.
Example
Gr24 = generalizedFlagVariety("A",3,{2}); --the Grassmannian of projective lines in projective 3-space
O1 = ampleKClass Gr24 -- the O(1) bundle on Gr24 as an equivariant K-class
E = O1 + (O1*O1)
peek E
SeeAlso
makeKClass
(symbol *, KClass, KClass)
///
doc ///
Key
generalizedFlagVariety
(generalizedFlagVariety, String, ZZ, List)
(generalizedFlagVariety, String, ZZ, List, Ring)
Headline
makes a generalized flag variety as a GKM variety
Usage
X = generalizedFlagVariety(LT,d,L)
X = generalizedFlagVariety(LT,d,L,R)
Inputs
LT:String
one of "A", "B", "C", or "D"
d:ZZ
the dimension of the root system
L:List
of integers strictly between 1 and d (inclusive)
R:Ring
the character ring of the torus acting on the generalized flag variety
Outputs
X:GKMVariety
representing the corresponding generalized flag variety
Description
Text
Let $G$ be the Lie group corresponding to $LT_d$, and
let $w = a_1w_1 + \cdots + a_dw_d$ be a nonnegative $\mathbb Z$-linear combination of fundamental weights
in the root system of type $LT_d$, where $a_i$ is the number of times $i$ appears in the list $L$.
(See @TO "Example: generalized flag varieties"@ for conventions regarding classical Lie groups and
their root systems).
This method outputs the GKM variety representing the generalized flag variety $G/P$ embedded in the irreducible
representation of $G$ with the highest weight $w$.
Text
The following example features the Lagrangian Grassmannian $LGr(2,4)$ of 2-dimensional subspaces
in $\mathbb C^4$ that are isotropic under the standard alternating form. Its @TO MomentGraph@ is a complete
graph on 4 vertices.
Example
LGr24 = generalizedFlagVariety("C",2,{2})
peek LGr24
momentGraph LGr24
euler ampleKClass LGr24
Caveat
Spin groups have not been implemented.
SeeAlso
"Example: generalized flag varieties"
flagMap
///
doc ///
Key
EquivariantMap
Headline
the class of all equivariant morphisms between GKM varieties
Description
Text
Given two GKM varieties $X$ and $Y$, an equivariant morphism from $X$ to $Y$
induces a map from the torus-fixed points of $X$ to the torus-fixed points of $Y$.
Text
A @TO "EquivariantMap"@ C is a @TO "HashTable"@
consisting of three keys:
@UL{
{TT "source", ", whose value is a ", TO "GKMVariety", " corresponding to the domain of f"},
{TT "target", ", whose value is a ", TO "GKMVariety", " corresponding to the codomain of f"},
{TT "ptsMap", ", whose value is a ", TO "HashTable", "; its keys are ", TT "X.points", " and the values are
points of ", TT "Y.points", " that the key maps to."}
}@
SeeAlso
(symbol **, EquivariantMap, EquivariantMap)
(compose, EquivariantMap, EquivariantMap)
(map, GKMVariety, GKMVariety, List)
flagMap
pullback
pushforward
(euler, KClass)
///
doc ///
Key
(symbol **, EquivariantMap, EquivariantMap)
Headline
computes the product of two equivariant morphisms
Usage
f ** g
Inputs
f:EquivariantMap
g:EquivariantMap
Outputs
:EquivariantMap
the product of f and g
Description
Text
This method computes the cartesian product of two equivariant morphisms.
Example
R = makeCharacterRing 3;
X = generalizedFlagVariety("A",2,{1,2},R);
Y = generalizedFlagVariety("A",2,{1},R);
f = flagMap(X,Y); -- the projection of Fl(1,2;3) onto Gr(2,3)
h = f ** f
peek h
SeeAlso
(map, GKMVariety, GKMVariety, List)
flagMap
(compose, EquivariantMap, EquivariantMap)
///
doc ///
Key
(compose, EquivariantMap, EquivariantMap)
Headline
computes the composition of two equivariant morphisms
Usage
compose(f,g)
Inputs
f:EquivariantMap
g:EquivariantMap
Outputs
:EquivariantMap
the composition of f and g
Description
Text
This method computes the composition of two equivariant morphisms. The
following example constructs the composition of two projection maps between
standard flag varieties.
Example
R = makeCharacterRing 4;
X = generalizedFlagVariety("A",3,{1,2,3},R);
Y = generalizedFlagVariety("A",3,{2,3},R);
Z = generalizedFlagVariety("A",3,{2},R);
f = flagMap(X,Y); --the projection of Fl(1,2,3;4) onto Fl(2,3;4)
g = flagMap(Y,Z); --the projection of Fl(2,3;4) onto Gr(2;4)
h = compose(g,f)
h === flagMap(X,Z)
SeeAlso
(symbol **, EquivariantMap, EquivariantMap)
(map, GKMVariety, GKMVariety, List)
flagMap
///
doc ///
Key
(map, GKMVariety, GKMVariety, List)
Headline
creates a EquivariantMap
Usage
f = map(X,Y,L)
Inputs
X:GKMVariety
the source GKM variety of the map
Y:GKMVariety
the target GKM variety of the map
L:List
of pairs (x,y) where x and y are members of @TT "X.points"@ and @TT "Y.points"@, respectively
Outputs
f:EquivariantMap
Description
Text
This method creates a @TO EquivariantMap@ given a GKM variety $X$, a GKM variety $Y$,
and a list @TT "L"@ of pairs (x,y) where x and y are members of @TT "X.points"@ and @TT "Y.points"@ (respectively),
indicating that the torus-fixed point x of X is sent to the torus-fixed point y of Y under the map.
Text
The following describes the projection from the third Hizerbruch surface to the projective
line.
Example
R = makeCharacterRing 2;
F3 = makeGKMVariety(hirzebruchSurface 3,R);
PP1 = projectiveSpace(1,R);
L = {({0,1},set {0}), ({0,3}, set{0}), ({1,2}, set{1}), ({2,3}, set{1})};
f = map(F3,PP1,L)
Caveat
This does not check that the morphism is well defined. In particular, it does not
verify that the map on torus-fixed points is induced by a morphism of GKM varieties.
SeeAlso
diagonalMap
flagMap
(pullback, EquivariantMap)
pushforward
(euler, KClass)
///
doc ///
Key
flagMap
(flagMap, GKMVariety, GKMVariety)
Headline
creates equivariant maps between generalized flag varieties
Usage
f = flagMap(X,Y)
Inputs
X:GKMVariety
the source generalized flag variety
Y:GKMVariety
the target generalized flag variety
Outputs
f:EquivariantMap
Description
Text
Let $L =\{k_1,\dots,k_m\}$ be a set of ranks of linear subscpaces of $\mathbb C^n$ and consider
a subset $L' \subseteq L$. Let $X = Fl(L; n)$ and $Y=Fl(L';n)$ be the associated generalized
flag varieties (if they exist). This method produces the canonical projection from $X$ to $Y$
that forgets the linear subspaces having ranks $L \setminus L'$.
Example
R = makeCharacterRing 3
X = generalizedFlagVariety("B",3,{1,2},R);
Y1 = generalizedFlagVariety("B",3,{2},R);
Y2 = generalizedFlagVariety("B",3,{1},R);
peek flagMap(X,Y1)
peek flagMap(X,Y2)
SeeAlso
(map, GKMVariety, GKMVariety, List)
diagonalMap
generalizedFlagVariety
///
doc ///
Key
(pullback, EquivariantMap)
Headline
computes the pullback map of equivariant K-classes of an equivariant map
Usage
pullback(f)
Inputs
f:EquivariantMap
Outputs
:FunctionClosure
whose input is a @TO KClass@ on the target @TO GKMVariety@ of f and output is its pullback along f
Description
Text
Given two GKM varieties $X$ and $Y$, this method computes the pullback of a @TO KClass@ on $Y$
along an equivariant morphism $X \to Y$.
Example
R = makeCharacterRing 4;
FlGr = generalizedFlagVariety("A",3,{1,2},R)
Gr24 = generalizedFlagVariety("A",3,{2},R)
f = flagMap(FlGr,Gr24)
O1 = ampleKClass Gr24
(pullback f)(O1)
SeeAlso
flagMap
pushforward
///
doc ///
Key
pushforward
(pushforward, EquivariantMap)
Headline
computes the pushforward map of equivariant K-classes of an equivariant map
Usage
pushforward(f)
Inputs
f:EquivariantMap
Outputs
:FunctionClosure
whose input is a @TO KClass@ on the source @TO GKMVariety@ of f and output is its pushforward along f
Description
Text
Given two GKM varieties $X$ and $Y$, this method computes the pushforward of a @TO KClass@ on $X$
along an equivariant morphism $X \to Y$.
Example
R = makeCharacterRing 4;
FlGr = generalizedFlagVariety("A",3,{1,2},R)
Gr24 = generalizedFlagVariety("A",3,{2},R)
f = flagMap(FlGr,Gr24)
O1 = ampleKClass FlGr
(pushforward f)(O1)
SeeAlso
flagMap
(pullback, EquivariantMap)
pushforward
(euler, KClass)
///
doc ///
Key
(euler, KClass)
Headline
computes the equivariant Euler characteristic of an equivariant K-class
Usage
euler C
Inputs
C:KClass
Outputs
:RingElement
in the character ring of the torus of the GKM variety on which C is defined
Description
Text
This method computes the pushforward of a @TO KClass@ on a @TO GKMVariety@ $X$ along the structure map
$X \to pt$, where $pt$ is a point with trivial torus-action.
Example
PP3 = projectiveSpace 3
O1 = ampleKClass PP3
euler O1
SeeAlso
pushforward
ampleKClass
///
doc ///
Key
projectiveSpace
(projectiveSpace, ZZ)
(projectiveSpace, ZZ, Ring)
Headline
constructs projective space as a GKM variety
Usage
projectiveSpace n
projectiveSpace(n,R)
Inputs
n:ZZ
R:Ring
Outputs
:GKMVariety
Description
Text
Given an integer $n$ this method constructs the n-dimensional projective space, $\mathbb P^n$, as a GKM variety. The action
of $(\mathbb C^*)^{n+1}$ on $\mathbb P^n$ is defined by
$(t_0, \ldots, t_n) \cdot (x_0, \ldots, x_n) = (t_0^{-1}x_0, \ldots, t_n^{-1}x_n)$.
Example
PP4 = projectiveSpace 4;
peek PP4
SeeAlso
flagMap
generalizedFlagVariety
///
doc ///
Key
makeCharacterRing
(makeCharacterRing, ZZ)
Headline
constructs the character ring of a torus
Usage
makeCharacterRing n
Inputs
n:ZZ
Outputs
:Ring
with n variables and inverses are allowed
Description
Text
Given an integer n, this method outputs the character ring of T = $(\mathbb C^*)^n$.
Example
R = makeCharacterRing 4
describe R
///
doc ///
Key
MomentGraph
Headline
the class of all moment graphs
Description
Text
The moment graph of a GKM variety $X$ with an action of a torus $T$ has vertices
corresponding to the $T$-fixed points $X^T$
and edges corresponding to the one-dimensional $T$-orbits. If $\{v_1,v_2\}$ is an edge and the
corresponding one-dimensional $T$-orbit closure is $\mathbb P^1$ where $v_1 = 0$ and $v_2 = \infty$,
then denote $m(v_1,v_2)$ to be the @EM "negative"@ of the character of the action of $T$ on
$\mathbb A^1 \subset \mathbb P^1$ (where $v_1 \in \mathbb A^1$).
Text
A @TO MomentGraph@ is a @TO HashTable@ with three keys:
@UL{
{TT "vertices", ", whose values represent the vertices of the moment graph"},
{TT "edges", ", whose value is a ", TO "HashTable", "; its keys are pairs {a,b} of elements in ",
TT "vertices", " representing the edges of the moment graph, and the values are the characters ", TEX "$m(a,b)$"},
{TT "HTpt", ", whose value is a ring representing the equivariant cohomology ring of a point"}
}@
Caveat
Functionalities concerning intersection cohomology of sheaves on moment graphs, which had been
implemented before (see @HREF{"https://people.math.umass.edu/~braden/MG/index.html","MG: moment graph computations"}@),
have not been imported into this package yet.
SeeAlso
momentGraph
makeGKMVariety
GKMVariety
///
doc ///
Key
momentGraph
(momentGraph, List, HashTable, Ring)
Headline
creates a moment graph
Usage
G = momentGraph(L,E,H)
Inputs
L:List
of vertices
E:HashTable
whose keys are lists of two vertices representing edges and values are characters of corresponding
1-dimensional orbits
H:Ring
a polynomial ring representing the equivariant cohomology ring of a point
Outputs
G:MomentGraph
Description
Text
This method creates a @TO MomentGraph@ from the data of vertices, edges and their associated characters,
and a ring representing the equivariant cohomology ring of a point (with trivial torus-action).
The following example is the moment graph of the projective 2-space $\mathbb P^2$.
Example
V = {set{0}, set{1}, set{2}};
E = hashTable {({set{0},set{1}},{-1,1,0}), ({set{0},set{2}},{-1,0,1}), ({set{1},set{2}},{0,-1,1})}
t = symbol t; H = QQ[t_0..t_2]
G = momentGraph(V,E,H)
peek G
underlyingGraph G
SeeAlso
MomentGraph
(underlyingGraph, MomentGraph)
(momentGraph, GKMVariety)
///
doc ///
Key
(momentGraph, GKMVariety)
Headline
view the moment graph of a GKM variety
Usage
G = momentGraph(X)
Inputs
X:GKMVariety
Outputs
G:MomentGraph
if a moment graph is defined for the @TO GKMVariety@ X
Description
Text
If a @TO MomentGraph@ has been defined for a @TO GKMVariety@ X, this method method returns the moment graph,
and returns error otherwise.
Example
momentGraph generalizedFlagVariety("A",3,{2})
SeeAlso
(momentGraph, GKMVariety, MomentGraph)
///
doc ///
Key
(momentGraph, GKMVariety, MomentGraph)
Headline
define a moment graph for a GKM variety
Usage
momentGraph(X,G)
Inputs
X:GKMVariety
G:MomentGraph
Outputs
:null
Description
Text
This methods sets a given @TO MomentGraph@ G to be the moment graph of a @TO GKMVariety@ X.
If a moment graph was already defined for X, then overwrites it and prints that it has done so.
Example
R = makeCharacterRing 4
X = makeGKMVariety({set{0},set{1},set{2},set{3}},R)
X.?momentGraph
PP3 = projectiveSpace 3
G = momentGraph PP3
momentGraph(X,G)
X.?momentGraph
momentGraph X
momentGraph(X,G)
SeeAlso
(momentGraph, GKMVariety)
(momentGraph, List, HashTable, Ring)
///
doc ///
Key
(underlyingGraph, MomentGraph)
Headline
the underlying (undirected) graph of a moment graph
Usage
underlyingGraph(G)
Inputs
G:MomentGraph
Outputs
:Graph
Description
Text
This method outputs the underlying undirected @TO Graph@ of a moment graph.
Example
G = momentGraph projectiveSpace 3
underlyingGraph G
SeeAlso
MomentGraph
///
doc ///
Key
(symbol **, MomentGraph, MomentGraph)
Headline
the product of two moment graphs
Usage
G ** H
Inputs
G:MomentGraph
H:MomentGraph
Outputs
:MomentGraph
Description
Text
Let $G$ and $H$ be moment graphs associated to the GKM varieties $X$ and $Y$, respectively. This function produces
the moment graph of $X ** Y$; the latter is a GKM variety via the diagonal action of the torus.
Example
G = momentGraph projectiveSpace 1;
H = momentGraph generalizedFlagVariety("C",2,{2}); -- The isotropic Grassmannian SpGr(2,4)
J = G ** H;
peek J
SeeAlso
MomentGraph
(symbol **, GKMVariety, GKMVariety)
///
doc ///
Key
ampleKClass
(ampleKClass, GKMVariety)
(ampleKClass, GKMVariety, KClass)
Headline
the class of an ample line bundle
Usage
ampleKClass(X)
Inputs
X:GKMVariety
C:KClass
Outputs
:KClass
Description
Text
If $X$ is a GKM variety with a distinguished ample equivariant line bundle, this method returns the @TO KClass@ of
the line bundle. If no such line bundle is defined, it allows the user to construct one.
Text
The following example describes the ample line bundle on the Lagrangian Grassmannian $SpGr(2,4)$. The line bundle
is precisely the pullback of O(1) under the Plücker embedding $SpGr(2,4) \to \mathbb P^4$.
Example
SpGr24 = generalizedFlagVariety("C",2,{2})
O1 = ampleKClass SpGr24
peek O1
SeeAlso
makeKClass
generalizedFlagVariety
///
doc ///
Key
orbitClosure
(orbitClosure, GKMVariety, Matrix)
[orbitClosure, RREFMethod]
Headline
computes the equivariant K-class of a torus orbit closure of a point in a generalized flag variety
Usage
C = orbitClosure(X,M)
Inputs
X:GKMVariety
M:Matrix
representing a point in a generalized flag variety
Outputs
C:KClass
Description
Text
Let $X$ be a generalized flag variety parameterizing flags of linear subspaces of dimensions $\{r_1, ... , r_k\}$
in $\mathbb C^n$ with $1 <= r_1 < \cdots < r_k$. Then a point $p$ of $X$ can be identified with a matrix $M$ of
size $r_k \times n$ such that the first $r_i$ rows of $M$ spans a subspace of dimension $r_i$. Given $X$ and
such a matrix $M$ representing the point $p$, this method computes the equivariant K-class of the closure of
the torus orbit of $p$.
Text
The following example computes the torus orbit closure of a point in the standard Grassmannian $Gr(2,4)$
and in the Lagrangian Grassmannian $SpGr(2,4)$.
Example
M = matrix(QQ,{{1,0,1,2},{0,1,2,1}})
X1 = generalizedFlagVariety("A",3,{2})
X2 = generalizedFlagVariety("C",2,{2})
C1 = orbitClosure(X1,M)
C2 = orbitClosure(X2,M)
peek C1
peek C2
Text
In type "A", the equivariant K-class of the orbit closure of a point coincides with that of its flag matroid.
Example
X = generalizedFlagVariety("A",3,{1,2})
Mat = random(QQ^2,QQ^4)
C = orbitClosure(X,Mat)
FM = flagMatroid(Mat,{1,2})
C' = makeKClass(X,FM)
C === C'
Text
In type "D", the orthogonal Grassmannian $SOGr(n,2n)$ has two connected components. To compute the
torus orbit closure of a point $p$ it suffices to restrict to either
$SOGr(n,n;2n)$ or $SOGr(n-1,n-1;2n)$, depending on which component $p$ is located in; see the last example
in @TO "Example: generalized flag varieties"@ for more details. Here is an example with $n=4$:
Example
R = makeCharacterRing 4
X1 = generalizedFlagVariety("D",4,{4,4},R)
X2 = generalizedFlagVariety("D",4,{3,3},R)
A = matrix{{1,3,-2,-1/4},{-1,-1,19,-61/4},{0,1,19,-73/4},{2,0,22,-89/4}};
B = matrix(QQ,{{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16}});
M = A | B
assert(A* transpose(B) + B *transpose(A) == 0) -- verifying that M is isotropic
C1 = orbitClosure(X1,M)
C2 = orbitClosure(X2,M)
peek C1
peek C2 -- since the point corresponding to M lies in X1, C2 is just the empty class i.e. 0
Text
By default the option RREFMethod is set to false. In this case the method produces the torus orbit closure by
only computing the minors of the matrix. If the option RREFMethod is set to true, the method row reduces
the matrix instead of computing its minors.
Example
X = generalizedFlagVariety("A",3,{1,2,3})
Mat = random(QQ^3,QQ^4)
time C = orbitClosure(X,Mat)
time C = orbitClosure(X,Mat, RREFMethod => true)
SeeAlso
generalizedFlagVariety
(makeKClass, GKMVariety, FlagMatroid)
///
doc ///
Key
cellOrder
(cellOrder, MomentGraph)
(cellOrder, GKMVariety)
Headline
the poset of a stratification of a GKM variety
Usage
P = cellOrder(G)
P = cellOrder(X)
Inputs
G:MomentGraph
on which a cell order has been defined already
X:GKMVariety
whose moment graph has a cell order defined already
Outputs
P:Poset
Description
Text
If a moment graph $G$ arises from a (possibly singular) GKM variety $X$ with an equivariant stratification,
with each strata having a unique torus-fixed point, the vertices of $G$
(which correspond to the torus-fixed point of $X$)
form a poset where $v_1 \leq v_2$ if the closure of the stratum corresponding to $v_1$ contains that of $v_2$.
The following example features the Schubert variety of projective lines in $\mathbb P^3$ meeting a distinguished line.
The poset of its stratification by smaller Schubert cells is a subposet of the Bruhat poset.
Example
Gr24 = generalizedFlagVariety("A",3,{2})
X = generalizedSchubertVariety(Gr24, {set{0,2}})
cellOrder X
SeeAlso
(cellOrder, MomentGraph, Poset)
bruhatOrder
///
doc ///
Key
(cellOrder, MomentGraph, Poset)
Headline
define a cell order on a moment graph
Usage
cellOrder(G,P)
Inputs
G:MomentGraph
P:Poset
whose ground set is the vertices of G
Description
Text
Defines a @TO Poset@ $P$ to be a cell order on the @TO MomentGraph@ $G$. Overwrites if there was
one already defined on $G$.
CannedExample
i2 : PP3 = projectiveSpace 3
o2 = a GKM variety with an action of a 4-dimensional torus
o2 : GKMVariety
i3 : cellOrder PP3
stdio:3:1:(3): error: no cell order defined on this moment graph
i4 : V = (momentGraph PP3).vertices
o4 = {set {0}, set {1}, set {2}, set {3}}
o4 : List
i5 : P = poset(V, {{V_0,V_1},{V_1,V_2},{V_2,V_3}})
o5 = P
o5 : Poset
i6 : cellOrder(momentGraph PP3, P)
i7 : cellOrder PP3
o7 = P
o7 : Poset
SeeAlso
cellOrder
bruhatOrder
///
doc ///
Key
bruhatOrder
(bruhatOrder, GKMVariety)
Headline
computes the Bruhat order on a generalized flag variety
Usage
P = bruhatOrder(X)
Inputs
X:GKMVariety
a generalized flag variety or Schubert variety
Outputs
P:Poset
the Bruhat order
Description
Text
Given a @TO GKMVariety@ $X$ created by either @TO generalizedFlagVariety@ or @TO generalizedSchubertVariety@,
computes and returns the Bruhat order corresponding to the Bruhat decomposition. The resulting poset is
cached in $X$, and can be accessed by either @TO cellOrder@ or @TO bruhatOrder@.
CannedExample
i1 : Fl3 = generalizedFlagVariety("A",2,{1,2})
o1 = a GKM variety with an action of a 3-dimensional torus
o1 : GKMVariety
i2 : cellOrder Fl3
stdio:2:1:(3): error: no cell order defined on this moment graph
i2 : P = bruhatOrder Fl3
o2 = P
o2 : Poset
i3 : #(coveringRelations P) == 8
o3 = true
i4 : cellOrder Fl3
o4 = P
o4 : Poset
SeeAlso
cellOrder
generalizedFlagVariety
generalizedSchubertVariety
///
doc ///
Key
generalizedSchubertVariety
(generalizedSchubertVariety, GKMVariety, Thing)
Headline
create a generalized Schubert variety
Usage
Y = generalizedSchubertVariety(X,pt)
Inputs
X:GKMVariety
obtained by @TO generalizedFlagVariety@
pt:Thing
an element in @TT "X.points"@
Outputs
Y:GKMVariety
the generalized Schubert variety corresponding to pt which is the union of all Bruhat cells
corresponding to elements in @TT "X.points"@ that are bigger than pt in the Bruhat order
Description
Text
This method creates a @TO GKMVariety@ that represent a generalized Schubert variety of a generalized flag variety.
The following example is the Schubert variety of projective lines in $\mathbb P^3$ meeting a distinguished point.
Example
X = generalizedSchubertVariety(generalizedFlagVariety("A",3,{2}),{set{0,3}})
SeeAlso
bruhatOrder
generalizedFlagVariety
///
doc ///
Key
charts
(charts, GKMVariety)
(charts, GKMVariety, List)
Headline
outputs the torus-invariant affine charts of a GKM variety
Usage
charts X
charts(X,L)
Inputs
X:GKMVariety
L:List
of lists
Outputs
H:HashTable
whose keys are the torus-fixed points of X and values are the (negatives) of characters of the
torus action on a torus-invariant affine chart around the corresponding point.
Description
Text
Assume $X$ is a GKM-variety for which there exists a contracting torus-invariant affine chart around each
torus-fixed point. For instance, generalized flag varieties and smooth toric varieties have this property.
This returns a @TO HashTable@ whose keys are the torus-fixed points of
$X$ and the values are the negatives of characters of the torus action on the associated
contracting affine chart.
Text
The following example describes the charts of the isotropic Grassmannian $SpGr(2,6)$.
Example
X = generalizedFlagVariety("C",3,{2});
X.charts
Text
If $X$ does not have its charts stored, we can manually cache it as follows.
Example
R = makeCharacterRing 2;
X = makeGKMVariety({{0,1},{0,3},{1,2},{2,3}},R);
peek X
L = {{{-1,0},{0,-1}},{{-1,0},{0,1}},{{-3,-1},{1,0}},{{1,0},{3,1}}};
charts(X,L);
peek X
peek makeGKMVariety hirzebruchSurface 3
SeeAlso
makeGKMVariety
generalizedFlagVariety
///
doc ///
Key
diagonalMap
(diagonalMap, GKMVariety)
Headline
constructs the diagonal morphism
Usage
diagonalMap X
Inputs
X:GKMVariety
Outputs
:EquivariantMap
Description
Text
Given a GKM variety $X$ this method constructs a @TO EquivariantMap@ representing the diagonal
morphism $X \to X \times X$. Note that $X \times X$ is a GKM variety via the diagonal action of the torus.
Example
X = generalizedFlagVariety("A",3,{2}); -- The Grassmannian Gr(2,4)
f = diagonalMap X;
peek f
SeeAlso
(map, GKMVariety, GKMVariety, List)
///
doc ///
Key
lieType
(lieType, GKMVariety)
Headline
outputs the Lie type of a generalized flag variety
Usage
lieType X
Inputs
X:GKMVariety
Outputs
:String
of the form "A", "B", "C" or "D"
Description
Text
If $X$ is a generalized flag variety or a generalized Schubert variety constructed using the method
@TO generalizedFlagVariety@ or @TO generalizedSchubertVariety@, this method outputs the Lie type of $X$.
Example
X = generalizedFlagVariety("A",3,{2}); -- The Grassmannian Gr(2,4)
Y = generalizedFlagVariety("B",2,{1}); -- The Orthogonal Grassmannian SOGr(1,5)
lieType(X)
lieType(Y)
Text
If the GKM variety is not a generalized flag variety or a generalized Schubert variety, prints error.
Caveat
The method @TO projectiveSpace@ dose not cache the Lie type.
SeeAlso
generalizedFlagVariety
///
doc ///
Key
affineToricRing
(affineToricRing, List)
(affineToricRing, Matrix)
Headline
computes the toric ring associated to a monomial map
Usage
affineToricRing L
affineToricRing M
Inputs
L:List
of lists; each list corresponds to a vector in $\mathbb Z^n$
M:Matrix
with each column corresponding to a vector in $\mathbb Z^n$
Outputs
:Ring
Description
Text
Given a list $\{v_1,...,v_d\}$ of vectors in $\mathbb Z^n$ this function computes
the toric ring $R/I$ where $R$ is the polynomial ring $\mathbb{Q}[x_1,\dots,x_d]$
with $x_i$ having degree $v_i$ and $I$ is the associated toric ideal. In particular
$I$ is the kernel of the map $R \to \mathbb{Q}[y_1,\dots,y_n]$ defined by
$x_i \mapsto \mathbb y^{v_i}$.
Example
L = {{2,0},{1,1},{0,2}};
X = affineToricRing L; -- The singular quadric in A^3
I = ideal X
hilbertSeries I
///
doc ///
Key
trivialKClass
(trivialKClass, GKMVariety)
Headline
the equivariant K-class of the structure sheaf
Usage
trivialKClass X
Inputs
X:GKMVariety
Outputs
:KClass
Description
Text
Given a GKM variety $X$ this function computes the @TO KClass@ of the
structure sheaf $O_X$. In terms of the localization map, $O_X$ corresponds
to the constant function $1$.
Example
X = projectiveSpace 3;
C = trivialKClass X;
peek C
///
doc ///
Key
FlagMatroid
Headline
the class of all flag matroids
Description
Text
A flag matroid $\mathbf M$ is an ordered list $\{M_1, \ldots, M_k\}$ of @TO Matroid@s on a common ground set
such that $M_i$ is a matroid quotient of $M_{i+1}$ for all $i=1, \ldots, k-1$. The matroids $M_i$'s are called
the "constituent" matroids of the flag matroid $\mathbf M$. The class @TO FlagMatroid@ is a @TO HashTable@ with
two keys:
@UL{
{TT "groundSet", ", whose value is a ", TO "Set", " representing the common ground set of the constituent matroids"},
{TT "constituents", ", whose value is a ", TO "List", " of ", TO "Matroid", "s"}
}@
Caveat
Flag matroids are the first examples beyond ordinary matroids of a more general combinatorial family
known as Coxeter matroids. Coxeter matroids have not been implemented yet.
SeeAlso
(flagMatroid, List)
(flagMatroid, Matrix, List)
(makeKClass, GKMVariety, FlagMatroid)
///
doc ///
Key
flagMatroid
(flagMatroid, List)
(flagMatroid, Matrix, List)
Headline
construct a flag matroid
Usage
FM = flagMatroid(ML)
FM = flagMatroid(A,L)
Inputs
ML:List
of @TO Matroid@s on a common ground set
A:Matrix
L:List
of integers between 1 and the number of rows of A (inclusive)
Outputs
FM:FlagMatroid
Description
Text
Given a list $ML$ of matroids on a common ground set, this method stores the data as a @TO FlagMatroid@.
Example
ML = {uniformMatroid(2,6),matroid completeGraph 4}
FM = flagMatroid(ML)
isWellDefined FM
Text
For $A$ an $r\times n$ matrix over a field and $L = \{r_1, \ldots, r_k}$ a list of integers,
let $M_i$ be the @TO Matroid@ defined by the columns of the matrix obtained by the first $r_i$ rows of $A$.
These matroids form a flag matroid $\mathbf M = \{M_1, \ldots, M_k\}$.
This method creates this @TO FlagMatroid@.
Example
A = random(QQ^2,QQ^4)
FM = flagMatroid(A,{1,2})
Caveat
When a list of matroids is given as input, this method does not check if the flag matroid is well-defined.
SeeAlso
(isWellDefined, FlagMatroid)
FlagMatroid
///
doc ///
Key
(isWellDefined, FlagMatroid)
Headline
check if a flag matroid is well-defined
Usage
isWellDefined(FM)
Inputs
FM:FlagMatroid
Outputs
:Boolean
Description
Text
A @TO FlagMatroid@ with constituent matroids $\{M_1, \ldots, M_k\}$ is well-defined if $M_i$ is
a matroid quotient of $M_{i+1}$ (i.e. every flat of $M_i$ is a flat of $M_{i+1}$) for all $i = 1, \ldots, k-1$.
Example
FM = flagMatroid {uniformMatroid(2,4),uniformMatroid(3,4)}
isWellDefined FM
FMbad = flagMatroid {uniformMatroid(2,4), uniformMatroid(1,2)++uniformMatroid(2,2)}
isWellDefined FMbad
SeeAlso
FlagMatroid
(flagMatroid, List)
///
doc ///
Key
(bases, FlagMatroid)
Headline
compute the bases of a flag matroid
Usage
B = bases(FM)
Inputs
FM:FlagMatroid
Outputs
B:List
Description
Text
An ordered list $\{B_1, \ldots, B_k\}$ of sets is a basis of a flag matroid $\mathbf M = \{M_1, \ldots, M_k\}$
if $B_i$ is a basis of $M_i$ and $B_i \subseteq B_{i+1}$ for all $i$. This method computes the bases of a
flag matroid.
Example
FM = flagMatroid {uniformMatroid(2,4),uniformMatroid(3,4)}
bases FM
SeeAlso
FlagMatroid
///
doc ///
Key
(latticePoints, FlagMatroid)
Headline
lattice points of a base polytope of a flag matroid
Usage
P = latticePoints(FM)
Inputs
FM:FlagMatroid
Outputs
P:List
of lists of integers representing the lattice points
Description
Text
For a basis $B= \{B_1, \ldots, B_k\}$ of a flag matroid $M$ (see @TO (bases, FlagMatroid)@),
let $e_B$ be the sum over $i = 1, \ldots, k$ of the indicator vectors of $B_i$.
The base polytope of a flag matroid $M$
is the convex hull of $e_B$ as $B$ ranges over all bases of $M$. This method computes the lattice points
of the base polytope of a flag matroid, exploiting the strong normality property as proven in [CDMS18].
Example
FM = flagMatroid {uniformMatroid(1,4),uniformMatroid(2,4)}
P = latticePoints FM
Text
In terms of equivariant K-theory, the lattice points of the base polytope of a flag matroid is
equal to the integer-point transform of the equivariant Euler characteristic (see @TO (euler, KClass)@) of the @TO KClass@
defined by the flag matroid shifted by the $O(1)$ bundle on the (partial) flag variety.
Example
X = generalizedFlagVariety("A",3,{1,2})
FM = flagMatroid {uniformMatroid(1,4),uniformMatroid(2,4)}
C = makeKClass(X,FM)
chiCO1 = euler(C * ampleKClass X)
set P === set exponents chiCO1
SeeAlso
(bases, FlagMatroid)
(euler, KClass)
///
doc ///
Key
(makeKClass, GKMVariety, FlagMatroid)
Headline
the equivariant K-class of a flag matroid
Usage
C = makeKClass(X,FM)
Inputs
X:GKMVariety
of @TO lieType@ "A" created by @TO generalizedFlagVariety@
FM:FlagMatroid
Outputs
C:KClass
Description
Text
A flag matroid of whose constituent matroids have ranks $r_1, \ldots, r_k$ and ground set size $n$ defines a
@TO KClass@ on the (partial) flag variety $Fl(r_1,\ldots, r_k;n)$. When the flag matroid arises
from a matrix representing a point on the (partial) flag variety, this equivariant K-class coincides with
that of the structure sheaf of its torus orbit closure.
See [CDMS18] or [DES20].
Example
X = generalizedFlagVariety("A",2,{1,2})
A = matrix{{1,2,3},{0,2,3}}
FM = flagMatroid(A,{1,2})
C1 = makeKClass(X,FM)
C2 = orbitClosure(X,A)
C1 === C2
SeeAlso
(latticePoints, FlagMatroid)
orbitClosure
flagGeomTuttePolynomial
///
doc ///
Key
flagGeomTuttePolynomial
(flagGeomTuttePolynomial, FlagMatroid)
Headline
computes the flag-geometric Tutte polynomial of flag matroids
Usage
flagGeomTuttePolynomial(FM)
Inputs
FM:FlagMatroid
Outputs
:RingElement
a polynomial in variables $x,y$
Description
Text
This method computes the flag-geometric Tutte polynomial of a @TO FlagMatroid@, defined via a push-pull of the
@TO KClass@ of the flag matroid. See Definition 6.1 of [DES20].
The following is the example 8.24 in [CDMS18].
Example
FM = flagMatroid {uniformMatroid(1,3),uniformMatroid(2,3)}
flagGeomTuttePolynomial FM
Text
The following example negatively answers Conjecture 9.2 of [CDMS18], which had conjectured that all coefficients
of the flag-geometric Tutte polynomial of a flag matroid are nonnegative.
CannedExample
i1 : FM = flagMatroid {uniformMatroid(1,5),uniformMatroid(3,5)}
o1 = a flag matroid with rank sequence {1, 3} on 5 elements
o1 : FlagMatroid
i2 : flagGeomTuttePolynomial FM
3 4 3 3 2 4 3 2 2 3 4 3 2 2 3 4 3 2 2
o2 = x y + x y + 2x y + x y - x y + 3x*y + x y + 6x y + 9x*y + 4y + x + 3x y + 3x*y +
---------------------------------------------------------------------------------------------
3
y
o2 : ZZ[x, y]
Text
Here is another counterexample but one where no constituent matroids have rank 1 or corank 1.
CannedExample
i1 : FM = flagMatroid {uniformMatroid(2,6),uniformMatroid(4,6)}
o1 = a flag matroid with rank sequence {2, 4} on 6 elements
o1 : FlagMatroid
i2 : time flagGeomTuttePolynomial FM -- used 691.322 seconds
4 4 4 3 3 4 4 2 3 3 2 4 4 3 2 2 3 4 4 3
o2 = x y + 2x y + 2x y + 3x y - 6x y + 3x y + 4x y + 18x y + 18x y + 4x*y + 5x + 14x y
--------------------------------------------------------------------------------------------
2 2 3 4 3 2 2 3
+ 18x y + 14x*y + 5y + 2x + 6x y + 6x*y + 2y
o2 : ZZ[x, y]
Text
When the flag matroid has a single constituent (i.e. is a matroid), it agrees with the usual Tutte polynomial.
Example
M = matroid graph{{a,b},{b,c},{c,a},{a,d}}
flagGeomTuttePolynomial flagMatroid {M}, tuttePolynomial M
Caveat
The computation often does not finish within a reasonable time (< 10 min) if the ground set is bigger than 5.
SeeAlso
FlagMatroid
(makeKClass, GKMVariety, FlagMatroid)
///
doc ///
Key
setIndicator
(setIndicator, Set, ZZ)
Headline
computes the signed indicator vector of an admissible set
Usage
setIndicator(T,n)
Inputs
T:Set
n:ZZ
Outputs
:List
corresponding to a vector in $\mathbb Z^n$
Description
Text
Let $S$ be a set consisting of elements $s$, where $s$ is either equal to $i$ or $i^*$ with $0 \leq i \leq n-1$.
The set $S$ is said to be admissible if for any integer $i$, not both $i$ and $i^{*}$ are contained in $S$.
This method produces the signed indicator vector of $S$. In particular, the @TO "setIndicator"@ of $S$ is
$\sum c_ie_i \in \mathbb Z^n$ where $c_i = 1$ if $i \in T$, $c_i = -1$ if $i^{*} \in T$ and $0$ otherwise.
Example
S1 = set{1,2,4,5};
S2 = set{1,"2*"};
setIndicator(S1,7)
setIndicator(S2,3)
Text
If the set is not admissible it produces an error.
CannedExample
i1 : S3 = set{1,"1*","2*",3}
o1 = set {1, 1*, 2*, 3}
o1 : Set
i2 : setIndicator(S3,4)
stdio:2:1:(3): error: the signed subset is not admissible
SeeAlso
generalizedFlagVariety
///
undocumented {
(net, GKMVariety),
(net, MomentGraph),
makeHTpt,
characterRing,
constituents,
KPolynomials,
HTpt,
points,
ptsMap,
RREFMethod,
tHilbNumer,
toCharacterRing,
toFraction,
(toFraction, RingElement, RingElement, Ring),
unastrsk,
(net, KClass),
(net, EquivariantMap),
(net, FlagMatroid),
signedPermutations
}
-*--documentation template
doc ///
Key
Headline
Usage
Inputs
Outputs
Description
Text
Blah
Example
X = 1
Caveat
SeeAlso
///
--*-