Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof...

Grothendieck expansions of symmetricpolynomials

Jason Bandlow (joint work with Jennifer Morse)

University of Pennsylvania

August 3rd, 2010 – FPSACSan Francisco State University

Outline

Symmetric functions

Tableaux–Schur expansions

Grothendieck functions and their dual basis

Main theorem

Examples

Sketch of proof

The monomial basis

The monomial symmetric functions are indexed by partitions

λ = (λ1, λ2, . . . , λk) λi ≥ λi+1

mλ =∑α

xα

α a rearrangement of the parts of λ and infinitely many 0’s

Example

m2,1 =(x21x2 + x1x2

2 ) + (x1x23 + x2

1x3) + . . .

+ (x22x3 + x2x2

3 ) + . . .

The complete homogeneous basis

The (complete) homogeneous symmetric functions are defined by

hi =∑λ`i

mλ

hλ = hλ1hλ2 . . . hλk

Example

h3 = m3 + m2,1 + m1,1,1

h4,2,1 = h4h2h1

The Hall inner product

Defined by

〈hλ,mµ〉 =

{1 if λ = µ

0 otherwise

Proposition

If {fλ}, {f ∗λ } and {gλ}, {g∗λ} are two pairs of dual bases with

fλ =∑µ

Mλ,µgµ

then

g∗µ =∑λ

Mλ,µf ∗λ

Semistandard Young tableaux

A left-and-bottom justified, partition-shaped array of numbers,weakly increasing across rows and strictly increasing up columns.

Example

7 75 6 6 83 3 4 71 1 2 2 2

Shape: (5, 4, 4, 2)Evaluation: (2, 3, 2, 1, 1, 2, 3, 1)Reading word: 775668334711222

Knuth equivalence

An equivalence relation on words generated by

yxz ≡ yzx if x < y ≤ z

xzy ≡ zxy if x ≤ y < z

Key Fact

Every word is Knuth equivalent to the reading word of exactly onetableau.

Example

rw

(3 41 2 3

)= 34123 ≡ 31423 ≡ 31243 ≡ 13243 ≡ 13423

The Schur basis

DefinitionThe Schur functions are given by

sλ =∑

T∈SSYT (λ)

xev(T )

Example

s2,1 = x21x2+ x1x2

2+ 2x1x2x3+ · · ·21 1

21 2

31 2

21 3 · · ·

Fact: Schur functions are a self-dual basis of the symmetricfunctions.

The Schur basis

Using the fact that Schur functions are symmetric, we can rewritethe definition as

sλ =∑µ

Kλ,µmµ

where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.

Using the proposition about dual bases, we get

hµ =∑λ

Kλ,µsλ

which can be rewritten as

hµ =∑T∈Tµ

ssh(T )

where Tµ is the set of all semistandard tableaux of evaluation µ.

The Schur basis

Using the fact that Schur functions are symmetric, we can rewritethe definition as

sλ =∑µ

Kλ,µmµ

where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.Using the proposition about dual bases, we get

hµ =∑λ

Kλ,µsλ

which can be rewritten as

hµ =∑T∈Tµ

ssh(T )

where Tµ is the set of all semistandard tableaux of evaluation µ.


Elements of a family {fα} of symmetric functions havetableaux–Schur expansions if there exist sets Tα of semistandardtableaux and weight functions wtα such that

fα =∑T∈Tα

wtα(T )ssh(T )

Goal: find appropriate sets and modifications of wt so that

fα =∑S∈Sα

wtα(S)Gsh(S)

fα =∑R∈Rα

wtα(R)gsh(R)

where G and g are, respectively, the Grothendieck anddual-Grothendieck functions.


Examples

I hµ =∑

T∈Tµ ssh(T )

I Hµ[X ; t] =∑

T∈Tµ tch(T )ssh(T )

I sλsµ =∑

T∈Tλ,µ ssh(T )

I Fσ =∑

T∈Tσ ssh(T )

I A(k)µ [X ; t] =

∑T∈Tk,µ

tch(T )ssh(T )

I Hµ[X ; 1, t] =∑

T∈T(1n)tchµ(T )ssh(T )

I Hµ[X ; q, t]?=∑

T∈T(1n)qaµ(T )tbµ(T )ssh(T )

Grothendieck polynomials

I Introduced by Lascoux-Schutzenberger (1982)

I Represent K -theory classes of structure sheaves of Schubertvarieties in GLn

I Given by elements of Z[[x1, . . . , xn]]

I Analogous to Schubert polynomials

I Sign-alternating by degree, equal to Schubert polynomial inbottom degree

Stable Grassmannian Grothendieck functions

Stable:

I Due to Fomin-Kirillov

I Limit as number of variables →∞

Grassmannian:

I Indexed by partitions

I Power series of symmetric functions

I Sign-alternating by degree in the Schur basis

I Analog of Schur functions

Stable Grassmannian Grothendieck functions

Stable:

I Due to Fomin-Kirillov

I Limit as number of variables →∞Grassmannian:

I Indexed by partitions

I Power series of symmetric functions

I Sign-alternating by degree in the Schur basis

I Analog of Schur functions

Grassmannian Grothendieck functions

Theorem (Buch)

Gλ =∑

S∈SVT (λ)

ε(S)xev(S)

where SVT (λ) is the set of set-valued tableaux of shape λ.

Example

G1 = s1 −s1,1 +s1,1,1 . . .

1 12 123 · · ·

23 234 · · ·...

...

Set-valued tableaux

Example

234 45

1 12 3

shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1)

ε(S) = (−1)|ev(S)|+|sh(S)| = 1

From

Gλ =∑

S∈SVT (λ)

ε(S)xev(S)

we seeGλ = sλ ± higher degree terms

Set-valued tableaux

We introduce the reading word of a set-valued tableau, defined by:

I Proceed row by row, from top to bottom

I In each row, first ignore the smallest element of each cell.Then read the remaining elements from right to left, and fromlargest to smallest within each cell.

I Read the smallest elements of each cell from left to right.

Example

234 45

1 12 3

has reading word: 543242113.

Dual Grothendieck functionsWe denote the dual basis to the Grothendieck by g

Theorem (Lam, Pylyavskyy)

gλ =∑

R∈RPP(λ)

xev(R)

where RPP(λ) is the set of reverse plane partitions of shape λ.

Example

g2,1 = s2,1 +s2

21 1

11 1

21 3

11 2

......

Reverse plane partitions

Example

2 32 2 41 2 21 1 1 1

shape(R) = (4, 3, 3, 2) evaluation(R) = (4, 3, 1, 1)

From

gλ =∑

R∈RPP(λ)

xev(R)

we seegλ = sλ ± lower degree terms

Dual Grothendieck functions

We define the reading word of a reverse plane partition to be asubsequence of the usual reading word, where we only take thebottommost occurence of every letter in each column.

Example

2 32 2 41 2 21 1 1 1

has reading word 324221111.

Generalizing tableaux–Schur expansions

Given any set Tα of semistandard tableaux, we definecorresponding sets Sα (respectively Rα) of set-valued tableaux(reverse plane partitions) defined by

S ∈ Sα ⇐⇒ rw(S) ≡ rw(T ) for some T ∈ Tα

R ∈ Rα ⇐⇒ rw(R) ≡ rw(T ) for some T ∈ Tα

Given a statistic wtα on Tα, we can extend it to reading words bywtα(w) = wtα(T ) if w ≡ rw(T ).

Examples

If T =

43 41 2 ∈ Tα, then

43 41 2

431 24

34 41 2

4 41 23

4

1 234

43412 43412 43412 44312 44312

are in Sα

and

43 41 2

4 41 31 2

4 43 41 2

44 41 31 2 · · ·

43412 44312 43412 44312

are in Rα. All of these will have the same weight.

Examples

If T =

43 41 2 ∈ Tα, then

43 41 2

431 24

34 41 2

4 41 23

4

1 234

43412 43412 43412 44312 44312

are in Sα and

43 41 2

4 41 31 2

4 43 41 2

44 41 31 2 · · ·

43412 44312 43412 44312

are in Rα. All of these will have the same weight.

Statement of main theorem

Theorem (B-Morse)

Let fα be a family of symmetric functions with

fα =∑T∈Tα

wtα(T )ssh(T ).

Then

fα =∑S∈Sα

ε(S)wtα(S)gsh(S)

and

fα =∑R∈Rα

wtα(R)Gsh(R).

Complete homogeneous functions

We have a tableaux-Schur expansion of the homogeneous functionsby

hµ =∑T∈Tµ

ssh(T )

where Tµ is the set of all tableaux of evaluation µ. Thus we have

hµ =∑S∈Sµ

ε(S)gsh(S) =∑R∈Rµ

Gsh(R)

where Sµ is the set of all set-valued tableaux of evaluation µ andRµ is the set of all reverse plane partitions of evaluation µ.

Example

h3,2 = s3,2 +s4,1 +s5

2 21 1 1

21 1 1 2 1 1 1 2 2

= g3,2 +g4,1 +g5

−g3,1 −g4

21 1 12 1 1 12 2

= G3,2 +G4,1 +G5

+2G3,2,1 +2G4,1,1 + · · ·22 21 1 1

21 21 1 1

221 1 1 2

211 1 1 2

Hall-Littlewood functions

The Hall-Littlewood functions form a basis for the symmetricfunctions over the field Q(t). Interpretations as

I Deformation of Weyl character formula

I Graded Sn-character of certain cohomology rings

I Representation theory of groups of matrices over finite fields

among others.

We will use the version typically denoted by Hµ[X ; t] or Q ′µ(x ; t) inthe literature.

Example

H1,1,1[X ; t] = s1,1,1 + (t2 + t)s2,1 + t3s3


Theorem (Lascoux-Schutzenberger)

Hµ[X ; t] =∑T∈Tµ

tch(T )ssh(T )

where charge is a non-negative integer statistic defined on wordsand constant on Knuth equivalence classes.

The charge statistic

Example

3 42 2 31 1 1 2

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 0 0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 1 0 0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 1 0 0 0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 1 0 0 0 0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 1 0 0 1 0 0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 1 0 0 1 0 0 0

3 4 2 2 3 1 1 1 2


Example

3 42 2 31 1 1 2

0 1 0 0 1 0 0 0 1

3 4 2 2 3 1 1 1 2

charge(T ) = 3


We have

Hµ[X ; t] =∑T∈Tµ

tch(T )ssh(T )

where Tµ is again the set of all tableaux of evaluation µ. Hence

Hµ[X ; t] =∑S∈Sµ

ε(S)tch(S)gsh(S)

=∑R∈Rµ

tch(R)Gsh(R)

where Sµ (resp. Rµ) is the set of set-valued tableaux (resp. reverseplane partitions) of evaluation µ and the charge of S or R is thecharge of the reading word.

Example

H1,1,1[X ; t] = s1,1,1 + (t + t2)s2,1 + t3s3

=g1,1,1 + (t + t2)g2,1 + t3g3

− 2g1,1 − (t + t2)g2 + g1

321

21 3

31 2 1 2 3

231

312 12 3 1 23 123

Littlewood-Richardson tableaux

One version of the Littlewood-Richardson rule states that

sλsµ =∑

T∈Tλ,µ

ssh(T )

where Tλ,µ is the set of tableaux:

I with evaluation (λ1, · · · , λ`(λ), µ1, · · · , µ`(µ))I which have the reverse lattice property with respect to the

letters 1, · · · , `(λ), and

I which have the reverse lattice property with respect to theletters `(λ) + 1, · · · , `(λ) + `(µ)


Example

T(2,1),(2,1) is

2 41 1 3 3

421 1 3 3

2 3 41 1 3

42 31 1 3

32 41 1 3

4321 1 3

3 42 31 1

432 31 1


Corollary

sλsµ =∑

S∈Sλ,µ

ε(S)gsh(S) =∑

R∈Rλ,µ

Gsh(R)

where Sλ,µ (resp. Rλ,µ) is the set of all set-valued tableaux (resp.reverse plane partitions)

I with evaluation (λ1, · · · , λ`(λ), µ1, · · · , µ`(µ))I which have the reverse lattice property with respect to the

letters 1, · · · , `(λ), and

I which have the reverse lattice property with respect to theletters `(λ) + 1, · · · , `(λ) + `(µ)

Grothendieck and Schur functions

Corollary

sλ =∑

S∈Sλ,∅

ε(S)gsh(S)

sλ =∑

R∈Rλ,∅

Gsh(S)

Duality gives expansions of G and g into Schur functions.A different form of these expansions was given by Lenart, in termsof combinatorial objects now called elegant fillings.

Sketch of proof

Given

fα =∑T∈Tα

wt(T )ssh(T )

=∑T∈Tα

wt(T )∑

T ′∈SSYT (sh(T ))

xev(T′)

we want to show

fα =∑S∈Sα

ε(S)wt(S)gsh(S)

=∑S∈Sα

ε(S)wt(S)∑

R∈RPP(sh(S))

xev(R)

The involution

Example

423 41 23 4

21 31 3 4

423 41 23 4

21 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

21 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

•21 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

2•1 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

21• 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

211 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

21• 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

2•1 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

•21 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

432 41 23 4

21 31 3 4

The involution

Example

423 41 23 4

21 31 3 4

423 41 23 4

21 31 3 4

Sage-Combinat meeting tonight

Sage’s mission:

“To create a viable high-quality and open-source alternative toMapleTM, MathematicaTM, MagmaTM, and MATLABTM”

...“and to foster a friendly community of users and developers”

Tonight, Thorton Hall, Room 326

I 7pm-8pm: Introduction to Sage and Sage-Combinat

I 8pm-10pm: Help on installation & getting startedBring your laptop!

I Design discussions

Thank you for your attention.

Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof...

Documents

Transcript of Grothendieck expansions of symmetric polynomialslinux.bucknell.edu/~pm040/Slides/Bandlow.pdf · gof...