Type A Symmetric Varieties and Schubert Polynomials

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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 Algebra-Geometry-Combinatorics 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
well-known algebro-combinatorial 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
n1
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 G-variety X is spherical if it contains a dense B-orbit,or
equivalently,if it has finitely many B-orbits.A subgroup H  G
is spherical if G=H is a spherical G-variety.
Examples
Grassmannian Gr(k;n):The B-orbits (Schubert cells) are
parameterized by strings of k 0’s and n k 1’s.
Complete flag variety G=B:The B-orbits (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 G-variety and let Y;Y
0
be B-orbit closures in X.
Definition
The weak order on X is the (ranked) partial order on the set of
B-orbit 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 multi-valued.
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
one-line 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 non-degenerate skew-symmetric bilinear
forms on C
n
.
B-orbits 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 multi-valued.
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 non-degenerate symmetric bilinear forms
on C
n
.
B-orbits 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 multi-valued.
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.
B-orbits 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 multi-valued.
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 B-orbit 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 B-orbit,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 one-line 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 B-orbit,
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 one-line 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 B-orbit.Let  be a string of p +’s and q ’s,
representing a closure of a minimal B-orbit.The set
W():= W
#
(
max
;) consists of all w 2 S
n
whose one-line
notation is obtained by the following process
find adjacent positions occupied by opposite signs and
place the larger position in the left-most available spot and
the smaller position in the right-most 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 G-orbit 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 B-orbit
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 B-orbit of GL
n
=Sp
n
in its
wonderful embedding,then
i

([Y ]) =
Y
1i<jni
(x
i
+x
j
):
If Y is the closure of the minimal B-orbit of GL
n
=O
n
in its
wonderful embedding,then
i

([Y ]) =
Y
1ijni
(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 B-orbits of GL
n
=GL
p
GL
q
,but
we have not yet found an interpretation in terms of
Schubert polynomial identities.
Formulae for the non-minimal B-orbits 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
Non-Example (?)
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 W-sets for any interval in Type A
symmetric varieties.
We can compute W-sets for wonderful embeddings of
Type A symmetric varieties.
Computation of W-sets 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 B-orbits 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 B-orbits 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 G-orbit?
Can,Joyce,Wyser
Type A Symmetric Varieties and Schubert Polynomials