A normalization formula for the Jack polynomials in ...homepage.univie.ac.at ›...

A normalization formula for the Jack polynomials in

superspace.

Yvan Le BorgneLaBRI, CNRS/Universite de Bordeaux

ESI, May 28th 2008

(joint work with Luc Lapointe and Philippe Nadeau)

Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.

Outline

A normalisation formula for the Jack polynomials (J(α)Λ )Λ in superspace involves

[θ1 . . . θmxΛ11 . . . xΛN

N ]∑

certain tableaux T of “shape” Λ

∑

σ∈SN

ǫ(σ)W1(T , σ).

Computational observations lead Desrosiers, Lapointe and Mathieu (2007) toconjecture a more compact expression for the coefficient of this double sum:

3α+5

2α+3

α+2

0+1

α+1

α+3

0+1

0+1 1

(3α+5)(2α+3)(α+2)(α+1)(α+3)

What are superspace andJack polynomials ?Where does the doublesum come from ?

Some combinatorics toprove the conjecture by asimplification of thedouble sum.


I. The context where the double sum appears.

[θ1 . . . θmxΛ11 . . . xΛN

N ]∑


∑

σ∈SN

ǫ(σ)W1(T , σ).


Partial references

Jack polynomials in Superspace, Desrosiers, Lapointe, and Mathieu,Commun. Math. Phys. (2003) 242 331-360,

Orthogonality of Jack polynomials in superspace, Desrosiers, Lapointe andMathieu, Advances in Mathematics (2007) 212 361-388.

A recursion and a combinatorial formula for Jack polynomials, Knop andSahi, Inventiones mathematicae (1997) 128 9-22,

A normalization formula for the Jack polynomials in superspace and an

identity on partition, Lapointe, Le Borgne, NadeauarXiv:math.CO.0803.4182.


Symmetric functions in superspace

The space is made up of polynomials in 2N variables xii∪θii satisfying

xixj = xjxi , xiθj = θjxi , θiθj = −θjθi ( =⇒ θ2i = 0).

Diagonal action of the symmetric group SN , Kσxiθixjθk ≡ xσ(i)θσ(i)xσ(j)θσ(k).

A polynomial P is symmetric iff ∀σ ∈ SN , KσP = P .

Description of the basis (mΛ)Λ of monomial symmetric functions indexed bysuperpartitions:

Monomials as weight of a singletableau T of shape the superpartitionΛ. (herexev(T )θev(T ) = x4

1θ1x32 x3

3 x4θ4θ5)

Symmetrization :

mΛ =∑

σ∈S7

Kσx41θ1x

32x3

3 x4θ4θ5

1111

222

333

4

1

4

5i

i

xi

θi


Jack polynomials in superspace

(J(α)Λ )Λ is another basis of symmetric functions parametrized by α ∈ R.

There are many characterizations of these polynomials:

(Coperator ) as simultaneous eigenfunctions of operators related to theCalogero-Moser-Sutherland Model that may involve Cherednik’s operators,

(Corthogonality ) by a triangular decomposition in the monomial basis andorthogonality with respect to different scalar products,

(Ctableau) as the symmetrization∑

σ∈SN

Kσθ1 . . . θmE(α)Λ

of non-symmetric polynomials E(α)Λ satisfying a recurrence that can be

interpreted in terms of tableaux [Knop, Sahi] :

E(α)Λ =

∑

T 0-admissible tableau of “shape” Λ

d(α)T xev(T ).

Generalized to superspace, see [Desrosiers, Lapointe, Mathieu, Vinet . . . ]Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.

A first explicit combinatorial formula for 〈〈J(α)Λ |J

(α)Ω 〉〉α

Let 〈〈.|.〉〉α be a scalar product used in a characterization of type (Corthogonality ).Combining (Corthogonality ) and (Coperator ) [DLM] showed that

〈〈J(α)Λ |J

(α)Ω 〉〉α = αf (Λ) cmin

Λ (α)

cminΛ′ (1/α)

δΛ,Ω

where

JΛ = cminΛ (α)mΛmin

+∑

Ω 6=Λmin

cΛ,Ω(α)mΩ.Λ Λmin

m

Using (Ctableau) and the symmetry of mΛminwe obtain

cminΛ (α) = [θ1 . . . θmxm−1

1 . . . x0mxm+1 . . . xN ]

∑

σ∈SN

Kσθ1 . . . θm

(

∑

T

d(α)T xev(T )

)


II. Combinatorics to simplify the double sum


A more direct description of cminΛ (α).

cminΛ (α) = [θ1 . . . θmxm−1

1 . . . x0mxm+1 . . . xN ]

∑

σ∈SN

Kσθ1 . . . θm

(

∑

T

d(α)T xev(T )

)

Kσθ1 . . . θm ∝ θ1 . . . θm

requires σ = σ1 σ2

whereσ1 ∈ S

1...mN

andσ2 ∈ S

m+1...NN .

Kσxev(T )∝ x

ev(Λmin)

constrainsthe distributionof labels in T .

11111

22333

3 4

5

5 66

7

788

9

9

10

101111

12

121313

14

14 1515

11

12

2

23

3

34

44

Λ Λmin Kσ−1Λmin Λ

σ1

σ2

ǫ(σ1)


0-admissible tableaux related to cminΛ (α)

A tableau T is 0-admissible if all labels in cells of the same column are differentand when a label occurs in two consecutive columns, the right occurence is notbelow the left one.

11333

4

5

6

7

8

9

10

11

12

13

14

15

1

2

3

4

Kσ−1Λmin Λ

ǫ(σ1) Each label defines the state of its cell:

freecommon

criticali

i

ii

i

i

iii

The weight d(α)T is a product of hooks (not

yet defined) related to critical cells.




111 2

3

333

44

5

56

6

7

7

889

9

10

10

111112

12

131314

14

1515

1

2

3

4

Kσ−1Λmin Λ


freecommon

criticali

i

ii

i

i

iii






11

111 2

33

3

3

333

444

5

56

6

7

7

889

9

10

10

111112

12

131314

14

1515

1

2

3

4

Kσ−1Λmin Λ


freecommon

criticali

i

ii

i

i

iii






11

111 2

33

3

3

333

444

5

56

6

7

7

889

9

10

10

1111111212

12

1313131414

14

151515

1

2

3

4

Kσ−1Λmin Λ


freecommon

criticali

i

ii

i

i

iii






11

111 2

33

3

3

333

444

5

5

5

6

66

7

7

7

8

88

9

9

9

10

10

10

1111111212

12

1313131414

14

151515

1

2

3

4

Kσ−1Λmin Λ


freecommon

criticali

i

ii

i

i

iii






11

111 2

33

3

3

333

444

5

5

5

6

66

7

7

7

8

88

9

9

9

10

10

10

1111111212

12

1313131414

14

151515

1

2

3

4

Kσ−1Λmin Λ


freecommon

critical

i




Killing configurations with free cells


Killing configurations with free cells

The sign of the two configurations are opposite because ǫ((4 7) σ1) = −ǫ(σ1).

The (not yet defined) remaining parts d(α)T of their weights are the same since

they only involve critical cells and Λ.


A configuration without free cells


An application of LGV lemma

The LGV lemma is based on a combinatorial interpretation of

det(Mi ,j)1≤i ,j≤m =∑

σ

ǫ(σ)m∏

j=1

Mj ,σ(j)

.

in terms of systems of paths.

X

configurations

ǫ(σ1) = det

i

j

1 ≤ i , j ≤ m

X

paths j → i

The weights of systems of paths where at least one vertex is shared by twopaths cancel in pairs that can be described by an involution. Thus it remainsonly the configurations without free cells.


Evaluation of the determinant (not detailed yet)

In a particular case,

det

i

j

1 ≤ i , j ≤ m

X

paths j → i




=det

i

j

1 ≤ i , j ≤ m

X

paths j → i




=det

i

j

1 ≤ i , j ≤ m

X

paths j → i

In the general case,


Sorting rows for the “hook” formula



i

j

11

11

1

11

11

α ααααα

dΛ((j , i)) = 6α + 9



1

1

1

1

1

1

1

α α

αα

1

i

j

11

11

1

11

11

α ααααα

dΛ((j , i)) = 6α + 9



1

1

1

1

1

1

1

α α

αα

1

1

1

11

i

j

11

11

1

11

11

α ααααα

dΛ((j , i)) = 6α + 9



i

j

11

11

1

11

11

α ααααα

dΛ((j , i)) = 6α + 9


The normalization formula

〈〈J(α)Λ |J

(α)Ω 〉〉α

= δΛ,Ωα||+||−(||2 )

= δΛ,Ωα||+||−(||2 )

3α+5

2α+3

α+2

0+1

α+1

α+3

0+1

0+1

5/α+43/α+31/α+21/α+1

0/α+1

3/α+21/α+1

1/α+1

(5/α+4)(3/α+3)(1/α+2)(1/α+1)3 (3/α+2)(3α+5)(2α+3)(α+2)(α+1)(α+3)


Back to the determinant : dΛ((j , i)) = aj + bi

We use parameters (aj) on rows and (bi ) on columns to decompose the hooks.

dΛ((i , j))

a(i , Λ)

b(j , Λ)

a(i , Λ)

b(Λi + 1, Λ)

A circle on row j and column i implies bi = 1 − aj .


Circles and relations bi = 1 − aj

a1a2a3a4a5

aN

b1

1 − a1

b3

1 − a2

In this example m = 5=⇒ det(a1, . . . a5, b1, . . . b4)We have two relevant relations:b2 = 1 − a1 and b4 = 1 − a2

=⇒ det(a1, . . . , a5, b1, b2)

More surprisingly det(a1, . . . , a5, b1, b2) =∏

1≤i<j≤m

(1 + aj − ai ).

Moreover we will treat (aj) and (bi ) not related by relevant relations as formalvariables independant of the shaded part of the superpartition.


A proof in two steps

Even if there is no relation between (ai ) and (bj ),det(a1, . . . , am, b1, . . . bm−1) =

∏

1≤i<j≤m(1 + aj − ai).

ii

jj

detdet

When we add a circle related to a relevant relation bi = 1 − aj , we alsorestrict the graph for the system of paths. These two modifications leavesinvariant the determinant if initialy it does not depend on (bi).


Shape Invariance proof

We suppose that the first sum depends only on (aj) so that it is not modifed bythe introduced relation b4 = 1 − a3.

∑

systems

∑

systems

∑

systems

We then obtain the equality of the two first terms by a cancellation (describedby an involution) of the third sum.

b4

a3

The eventual horizontal stepis weighted by

a3 + b4 = 1

due to the introduced relation.


Recursive computation for the full square

P[k]j ,i =

∑

path p from j to i intersecting all rows j − 1, . . . j − k

w(p).

∆[k]j ,i ≡ P

[k]j ,i − P

[k]j−1,i = (aj + 1 − aj−k−1)P

[k+1]j ,i .

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

P[0]1,1 P

[0]1,2 P

[0]1,3 P

[0]1,4

P[0]2,1 P

[0]2,2 P

[0]2,3 P

[0]2,4

P[0]3,1 P

[0]3,2 P

[0]3,3 P

[0]3,4

P[0]4,1 P

[0]4,2 P

[0]4,3 P

[0]4,4

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

P[0]1,1 P

[0]1,2 P

[0]1,3 P

[0]1,4

P[1]2,1 P

[1]2,2 P

[1]2,3 P

[1]2,4

P[1]3,1 P

[1]3,2 P

[1]3,3 P

[1]3,4

P[1]4,1 P

[1]4,2 P

[1]4,3 P

[1]4,4

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

P[0]1,1 P

[0]1,2 P

[0]1,3 P

[0]1,4

∆[0]2,1 ∆

[0]2,2 ∆

[0]2,3 ∆

[0]2,4

∆[0]3,1 ∆

[0]3,2 ∆

[0]3,3 ∆

[0]3,4

∆[0]4,1 ∆

[0]4,2 ∆

[0]4,3 ∆

[0]4,4

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

P[0]1,1 P

[0]1,2 P

[0]1,3 P

[0]1,4

P[1]2,1 P

[1]2,2 P

[1]2,3 P

[1]2,4

P[2]3,1 P

[2]3,2 P

[2]3,3 P

[2]3,4

P[3]4,1 P

[3]4,2 P

[3]4,3 P

[3]4,4

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣


A double-counting for the factorisation’s recurrence

∆[k]j ,i ≡ P

[k]j ,i − P

[k]j−1,i = (aj + 1 − aj−k−1)P

[k+1]j ,i

is equivalent to

(aj − aj−k−1)P[k+1]j ,i =

(

P[k]j ,i − P

[k+1]j ,i

)

− P[k]j−1,i

where both expressions are the weighted sum of “extended” paths:

ii i

jj j

k + 1k + 1 k + 1

×(+1) ×(−1)


Conclusion

Using LGV lemma, we identified a determinant in the double sum

[θ1 . . . θmxΛ11 . . . xΛN

N ]∑


∑

σ∈SN

ǫ(σ)W1(T , σ).

We computed this determinant and then proved the following conjecturedexpression for the double sum:

3α+5

2α+3

α+2

0+1

α+1

α+3

0+1

0+1 1

(3α+5)(2α+3)(α+2)(α+1)(α+3)

This lead to a compacted normalization formula for Jack polynomials insuperspace involving product of certain hooks of the superpartition.


Appendix. Some skipped definitions

The Hamiltonian of the supersymmetric form of the trigonometricCalogero-Moser-Sutherland model seems to be

H =∑

i

(xi∂xi)2 +

1

α

∑

i<j

xi + xj

xi − xj(xi∂xi

−xj∂xj)−

2

α

∑

i<j

xixj

(xi − xj)2(1−K(ij)).

The scalar product for our normalization formula is

〈〈pΛ|pΩ〉〉α = (−1)m(m−1)/2αl(Λ)zΛsδΛ,Ω

where Λ = (Λ1, . . . ,Λm; Λs) and pΛ := pΛ1. . . pΛm

pΛm+1. . . pΛN

.

We are not using the following compatible scalar product

< A(x, θ)|B(x, θ) >α,N=Y

1≤j≤N

1

2πi

I

dxj

xj

Z

dθj θj

Y

1≤k 6=l≤N

1 −xk

xl

!1/α

A(x, θ)B(x, θ)

where xj = 1/xj and θi1 . . . θimθi1 . . . θim = 1.


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