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

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 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

1i<jni

(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

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 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

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