Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Type A Symmetric Varieties
and Schubert Polynomials
Michael Joyce (Tulane)
Mahir Can (Tulane) Ben Wyser (Illinois)
UIUC AlgebraGeometryCombinatorics Seminar
Tuesday,September 25,2012
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Motivation
Find combinatorial descriptions of the geometry of spherical
varieties.Apply these descriptions to better understand
wellknown algebrocombinatorial objects,e.g.Schubert
polynomials.
1
Weak Order on Spherical Varieties
2
Type A Symmetric Varieties
3
Results
4
Application to Schubert Polynomials
5
Extensions and Open Problems
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Notation
All groups and varieties are deﬁned over C.
G = GL
n
B =
8
>
>
<
>
>
:
0
B
B
@
0
0 0
0 0 0
1
C
C
A
9
>
>
=
>
>
;
G (Borel subgroup)
T =
8
>
>
<
>
>
:
0
B
B
@
0 0 0
0 0 0
0 0 0
0 0 0
1
C
C
A
9
>
>
=
>
>
;
G (maximal torus)
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Notation (cont.)
W = S
n
= symmetric group on n letters (Weyl group of G)
S = fs
1
;:::;s
n1
g,the simple transpositions of S
n
s
i
interchanges i and i +1,and ﬁxes everything else.
P
i
=
8
>
>
<
>
>
:
0
B
B
@
0
0
0 0 0
1
C
C
A
9
>
>
=
>
>
;
G (minimal parabolic subgroup)
The extra entry occurs in position (i +1;i).
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Spherical Varieties
Deﬁnition
A Gvariety X is spherical if it contains a dense Borbit,or
equivalently,if it has ﬁnitely many Borbits.A subgroup H G
is spherical if G=H is a spherical Gvariety.
Examples
Grassmannian Gr(k;n):The Borbits (Schubert cells) are
parameterized by strings of k 0’s and n k 1’s.
Complete ﬂag variety G=B:The Borbits (Schubert cells)
are parameterized by permutations in S
n
.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Weak Order
Let X be a Gvariety and let Y;Y
0
be Borbit closures in X.
Deﬁnition
The weak order on X is the (ranked) partial order on the set of
Borbit closures in X that is generated by the covering
relations,denoted Y = s
i
"Y
0
or
Y
Y
0
i
,
whenever Y = P
i
Y
0
and dimY = dimY
0
+1.
If P
i
Y
0
= Y
0
,then we set s
i
"Y
0
= Y
0
.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
The ‘Down’ Action
Deﬁnition
s
i
#Y =
(
Y if @Y
0
:Y = P
i
Y
0
and dimY = dimY
0
+1;
Y
0
if Y = P
i
Y
0
and dimY = dimY
0
+1:
Warning:The ‘function’ s
i
#Y may be multivalued.
Deﬁnition
Let w = s
i
1
s
i
2
s
i
l
be a reduced decomposition of w 2 S
n
.
w#Y = s
i
1
#(s
i
2
#( (s
i
l
#Y )))
This is independent of the choice of reduced decomposition.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Examples
Grassmannian Gr(k;n)
Weak order is the same as inclusion order.
Covering relations correspond to switching 01 to 10 in the
(i;i +1) positions of the string.
Complete ﬂag variety G=B
Weak order is strictly weaker than inclusion order.
Covering relations correspond to switching i and i +1 in the
oneline notation of w when i comes before i +1.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
GL
n
=Sp
n
GL
n
=O
n
GL
n
=GL
p
GL
q
GL
n
=Sp
n
(n = 2k even)
Parameterizes nondegenerate skewsymmetric bilinear
forms on C
n
.
Borbits are parameterized by the set of ﬁxed point free
involutions in S
n
.
Covering relations:
0
= s
i
# iff
1
(i) <
1
(i +1) and
0
= s
i
s
1
i
s
i
# is not multivalued.
Examples:
s
3
#(13)(27)(45)(68) = (14)(27)(35)(68)
s
5
#(17)(24)(38)(56) = (17)(24)(38)(56)
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
GL
n
=Sp
n
GL
n
=O
n
GL
n
=GL
p
GL
q
Example (GL
4
=Sp
4
)
(12)(34)
(13)(24)
(14)(23)
2
1;3
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
GL
n
=Sp
n
GL
n
=O
n
GL
n
=GL
p
GL
q
GL
n
=O
n
Parameterizes nondegenerate symmetric bilinear forms
on C
n
.
Borbits are parameterized by the set of involutions in S
n
.
Covering relations come in two types:
0
= s
i
# iff
1
(i) <
1
(i +1) and
0
= s
i
s
1
i
,or
ﬁxes i and i +1 and
0
= s
i
.
s
i
# is not multivalued.
Examples:
s
3
#(13)(27) = (14)(27)
s
5
#(13)(27) = (13)(27)(56)
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
GL
n
=Sp
n
GL
n
=O
n
GL
n
=GL
p
GL
q
Example (GL
3
=O
3
)
id
(12)
(23)
(13)
1
2
2
1
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
GL
n
=Sp
n
GL
n
=O
n
GL
n
=GL
p
GL
q
GL
n
=GL
p
GL
q
(p +q = n)
Parameterizes decompositions C
n
= U V with
dimU = p,dimV = q.
Borbits are parameterized by the set of involutions in S
n
with signs attached to the ﬁxed points such that there are
p q more +’s than ’s.
Covering relations comes in two types:
0
= s
i
# iff
1
(i +1) <
1
(i) and
0
= s
i
s
1
i
,or
interchanges i and i +1 and
0
= s
i
.
Signs of the common ﬁxed points must be the same.
s
i
# is multivalued.
Examples:
s
2
#(1
+
)(24)(3
+
) = (1
+
)(2
+
)(34)
s
2
#(1
+
)(23)(4
+
) = (1
+
)(2
+
)(3
)(4
+
);(1
+
)(2
)(3
+
)(4
+
)
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
GL
n
=Sp
n
GL
n
=O
n
GL
n
=GL
p
GL
q
Example (GL
3
=GL
2
GL
1
)
(13)(2
+
)
(1
+
)(23)
(12)(3
+
)
(1
+
)(2
+
)(3
)
(1
+
)(2
)(3
+
)
(1
)(2
+
)(3
+
)
1
2
2
2
1
1
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Standard formfor involutions
Deﬁnition
The standard form of an involution is its expression
= (a
1
;b
1
)(a
2
;b
2
) (a
k
;b
k
)
as a product of disjoint transpositions such that
a
i
< b
i
for all 1 i k and
a
1
< a
2
< < a
k
.
Remark:We let c
1
< c
2
< < c
l
denoted the ﬁxed points of .
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Maximal chains for weak order
Deﬁnition
W
#
(;
0
) = fw 2 S
n
:
0
= w# and`(w) = L(
0
) L()g;
where L() denotes the codimension of the Borbit closure
associated to in the given symmetric variety.
Let = (a
1
;b
1
)(a
2
;b
2
) (a
k
;b
k
) be an involution in standard
form.
For GL
n
=Sp
n
,L() =#inv(a
1
;b
1
;a
2
;b
2
;:::;a
k
;b
k
).
For GL
n
=O
n
,L() =#inv(a
1
;b
1
;a
2
;b
2
;:::;a
k
;b
k
) +k.
For GL
n
=GL
p
GL
q
(p q),
L() = C(p;q) #inv(a
1
;b
1
;a
2
;b
2
;:::;a
k
;b
k
) k.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Results for GL
n
=Sp
n
Theorem (CJW)
Let
max
denote the ﬁxed point free involution corresponding to
the dense Borbit,i.e.
max
= (1;2)(3;4) (n 1;n).Let
= (a
1
;b
1
)(a
2
;b
2
) (a
k
;b
k
) be a ﬁxed point free involution
written in standard form.The set W():= W
#
(
max
;) consists
of all w 2 S
n
such that in the oneline notation of w,
there are k “blocks” a
i
b
i
(i.e.a
i
is immediately left of b
i
);
for i < j,if b
i
< b
j
then the block a
i
b
i
must appear to the
left of a
j
b
j
,while if b
i
> b
j
the blocks may appear in either
relative order.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Examples in GL
6
=Sp
6
Examples
W((1;5)(2;6)(3;4)) = f152634;153426;341526g
W((1;6)(2;5)(3;4)) =
f162534;163425;251634;253416;341625;342516g
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Results for GL
n
=O
n
Theorem (CJW)
Let
max
be the involution corresponding to the dense Borbit,
i.e.
max
= id.Let = (a
1
;b
1
)(a
2
;b
2
) (a
k
;b
k
) be an involution
written in standard form.The set W():= W
#
(
max
;) consists
of all w 2 S
n
such that in the oneline notation of w,
b
i
appears to the left of a
i
and no number placed between
b
i
and a
i
has value between a
i
and b
i
;
for i < j,if b
i
< b
j
then a
i
must appear to the left of b
j
;
if a ﬁxed point is less than a
i
,it must be left of b
i
and if it is
greater than b
i
it must be right of a
i
;
the ﬁxed points must appear in increasing order.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Examples in GL
6
=O
6
Examples
W((1;4)(2;6)) = f341562;341625;413562;413625;416235g
W((1;5)(2;4)) =
f342516;345126;351426;423516;425136;451236;513426;514236g
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Results for GL
n
=GL
p
GL
q
Theorem (CJW)
Assume p q.Let
max
denote the involution corresponding to
the dense Borbit.Let be a string of p +’s and q ’s,
representing a closure of a minimal Borbit.The set
W():= W
#
(
max
;) consists of all w 2 S
n
whose oneline
notation is obtained by the following process
ﬁnd adjacent positions occupied by opposite signs and
place the larger position in the leftmost available spot and
the smaller position in the rightmost available spot;
cross out the two positions just used and repeat the
process;
after q steps,list the remaining positions in increasing
order.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Results for GL
n
=Sp
n
Results for GL
n
=O
n
Results for GL
n
=GL
p
GL
q
Examples in GL
6
=GL
4
GL
2
Examples
W(++++) = f561234g
W(++++) = f263451;361452;623415;631425g
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Schubert polynomials
Let x
i
denote the Chern class of the dual of the i
th
tautological
bundle on G=B.A classical result of Borel states that
H
(G=B;Z) = Z[x
1
;:::;x
n
]=I;
where I is the ideal generated by symmetric polynomials of
positive degree.
Deﬁnition
Let w 2 S
n
.The Schubert polynomial S
w
(x
1
;:::;x
n
) is the
unique polynomial in the span of monomials x
a
1
1
x
a
2
2
x
a
n
n
with
each a
i
n i that represents the cohomology class
[
Bw
0
wB=B] 2 H
(G=B;Z).
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Cohomological interpretation
Theorem (M.Brion)
Let i:G=P,!X be the inclusion of the closed Gorbit of the
wonderful embedding X of a Type A symmetric variety,and
i
:H
(X;Z)!H
(G=P;Z) the corresponding restriction map
on cohomology.If Y denotes the closure of the Borbit
containing the involution ,then
i
([Y ]) =
X
w2W
#
(
max
;)
S
w
1(x
1
;:::;x
n
):
Remark:Actually,this formula is correctly only up to a factor of
a power of 2 that is irrelevant for our application (and only
arises in the case GL
n
=O
n
).
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Factorization formulae
Theorem (B.Wyser)
If Y is the closure of the minimal Borbit of GL
n
=Sp
n
in its
wonderful embedding,then
i
([Y ]) =
Y
1i<jni
(x
i
+x
j
):
If Y is the closure of the minimal Borbit of GL
n
=O
n
in its
wonderful embedding,then
i
([Y ]) =
Y
1ijni
(x
i
+x
j
):
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Remarks
Together,the previous two slides write a sum of certain
Schubert polynomials as a product of linear factors.
Wyser’s computation of i
([Y ]) uses the localization
theorem in equivariant cohomology.
Wyser also has factorization formulae for the classes of the
closures of the minimal Borbits of GL
n
=GL
p
GL
q
,but
we have not yet found an interpretation in terms of
Schubert polynomial identities.
Formulae for the nonminimal Borbits can be obtained by
applying divided difference operators.It is not hard to keep
track of when such operators preserve a factorization
(canceling one term in the process).
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Example 1
Example (GL
4
=Sp
4
)
W
#
((12)(34);(14)(23)) = f1423;2314g
W
"
((14)(23);(12)(34)) = f1342;3124g
S
1342
+S
3124
= (x
1
+x
2
)(x
1
+x
3
)
(x
1
x
2
+x
1
x
3
+x
2
x
3
) +x
2
1
= (x
1
+x
2
)(x
1
+x
3
)
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Example 2
Example (GL
4
=O
4
)
W
#
(id;(14)(23)) = f4132;3412;3241g
W
"
((14)(23);id) = f2431;3412;4213g
S
2431
+ S
3412
+ S
4213
= x
1
x
2
(x
1
+x
2
)(x
1
+x
3
)
(x
2
1
x
2
x
3
+x
1
x
2
2
x
3
) + x
2
1
x
2
2
+ x
3
1
x
2
= x
1
x
2
(x
1
+x
2
)(x
1
+x
3
)
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
NonExample (?)
Example (GL
4
=GL
2
GL
2
)
W
#
((14)(23);(1
+
)(2
)(3
)(4
+
)) = f2431;4213g
W
"
((1
+
)(2
)(3
)(4
+
);(14)(23)) = f4132;3241g
S
4132
+ S
3241
= x
2
1
(x
1
x
2
+x
1
x
3
+x
2
x
3
)
(x
3
1
x
2
+x
3
1
x
3
) +x
2
1
x
2
x
3
= x
2
1
(x
1
x
2
+x
1
x
3
+x
2
x
3
)
Remark:One of Wyser’s factorization formulae yields
S
4132
+S
3241
x
2
2
x
2
3
(mod I);
where I is the ideal generated by symmetric polynomials of
positive degree.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Extensions
We can compute Wsets for any interval in Type A
symmetric varieties.
We can compute Wsets for wonderful embeddings of
Type A symmetric varieties.
Computation of Wsets for symmetric varieties of other
types should be fairly straightforward (use combinatorics of
twisted involutions developed by Richardson,Springer,
Hultman,etc.).
Wyser has found factorization formulas for (equivariant)
cohomology classes of closures of minimal Borbits for all
symmetric varieties of classical type.
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
Weak Order on Spherical Varieties
Type A Symmetric Varieties
Results
Application to Schubert Polynomials
Extensions and Open Problems
Open Problems
Is there a type independent version of our results for all
symmetric varieties in terms of root data?
How do you parameterize the Borbits and the weak order
for a general spherical variety?(Use combinatorics of
Luna’s spherical systems.)
Is there a geometric interpretation of the factorization
formulae?
Are there similar factorizations in the cohomology ring of
the wonderful embedding or only when restricted to the
closed Gorbit?
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials
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