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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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
I. The context where the double sum appears.
[θ1 . . . θmxΛ11 . . . xΛN
N ]∑
certain tableaux T of “shape” Λ
∑
σ∈SN
ǫ(σ)W1(T , σ).
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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 )
)
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
II. Combinatorics to simplify the double sum
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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)
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
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
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
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
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
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
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
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
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
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
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
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
ǫ(σ1) Each label defines the state of its cell:
freecommon
critical
i
The weight d(α)T is a product of hooks (not
yet defined) related to critical cells.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Killing configurations with free cells
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Killing configurations with free cells
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
A configuration without free cells
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
A configuration without free cells
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
A configuration without free cells
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
A configuration without free cells
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Evaluation of the determinant (not detailed yet)
In a particular case,
det
i
j
1 ≤ i , j ≤ m
X
paths j → i
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Evaluation of the determinant (not detailed yet)
In a particular case,
=det
i
j
1 ≤ i , j ≤ m
X
paths j → i
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Evaluation of the determinant (not detailed yet)
In a particular case,
=det
i
j
1 ≤ i , j ≤ m
X
paths j → i
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Evaluation of the determinant (not detailed yet)
In a particular case,
=det
i
j
1 ≤ i , j ≤ m
X
paths j → i
In the general case,
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Sorting rows for the “hook” formula
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Sorting rows for the “hook” formula
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Sorting rows for the “hook” formula
i
j
11
11
1
11
11
α ααααα
dΛ((j , i)) = 6α + 9
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Sorting rows for the “hook” formula
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
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Sorting rows for the “hook” formula
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
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Sorting rows for the “hook” formula
i
j
11
11
1
11
11
α ααααα
dΛ((j , i)) = 6α + 9
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Sorting rows for the “hook” formula
i
j
11
11
1
11
11
α ααααα
dΛ((j , i)) = 6α + 9
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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)
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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 .
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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).
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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)
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
Conclusion
Using LGV lemma, we identified a determinant in the double sum
[θ1 . . . θmxΛ11 . . . xΛN
N ]∑
certain tableaux T of “shape” Λ
∑
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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.
Yvan Le Borgne LaBRI, CNRS/Universite de Bordeaux A normalization formula for the Jack polynomials in superspace.
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