Row and column length restrictions of some classical
Schur function identities and the connection with
Howe dual pairs
Ronald C King, University of Southampton, UK
Inspired by a conjecture by Joris Van der Jeugt (University of Gent,
Belgium) including joint work with Angele Hamel (Wilfred Laurier University,
Canada)
8th International Conference on Combinatorics and Representation Theory
Graduate School of Mathematics, Nagoya University
Nagoya, Japan. September 2008
Nagoya - 2008
– p. 1/88
Motivation
Received query from Joris Van der Jeugt
(working with Stijn Lievens and Neli Stoilova)
Studying representations of the orthosymplectic Lie
superalgebra osp(1|2n) built using parabosons
Identified Fock space modules V (p) for any p ∈ N
Constructed unitary irreducible infinite-dimensional
representations V (p) = V (p)/M (p) where M (p) is the
maximal submodule of V (p), and found that
for p ≥ n irrep V (p) = V (p)
for p < n irrep V (p) = V (p)/M (p)
Also calculated the characters of both V (p) and V (p)
Nagoya - 2008
– p. 2/88
Van der Jeugt’s conjecture
Proposition [Van der Jeugt, Lievens and Stoilova, 2007]
Let x = (x1 , x2 , . . . , xn ), then
ch V (p) = (x1 x2 · · · xn )
p/2
X
sλ (x)
λ:ℓ(λ)≤p
Conjecture [Van der Jeugt, Lievens and Stoilova, 2007]
P
cη
X
sη (x)
(−1)
η
Q
sλ (x) = Q
1≤i≤n (1 − xi )
1≤j<k≤n (1 − xi xj )
λ:ℓ(λ)≤p
with the sum over all
notation
partitions η which in Frobenius
a1
a2 · · · ar
take the form η =
a1 + p a2 + p · · · ar + p
Nagoya - 2008
– p. 3/88
Macdonald’s Theorem
Joris Van der Jeugt asked if the result was known
If so where could it be found, if not could I supply a proof?
Angele Hamel reminded me of:
Theorem [Macdonald 79]
n−j
n+p+j−1
x
X
− xi
i
Q
sλ (x) = Q
1≤i≤n (1 − xi )
1≤j<k≤n (xj − xk )(1 − xj xk )
′
λ:ℓ(λ )≤p
Need to compare this with an immediate Corollary to Van
der Jeugt’s Conjecture
P
cη
X
(−1)
sη′ (x)
η
Q
sλ (x) = Q
1≤i≤n (1 − xi )
1≤j<k≤n (1 − xi xj )
′
λ:ℓ(λ )≤p
Nagoya - 2008
– p. 4/88
Strategy
Try to recast the numerator of Macdonald’s formula as a
signed sum of Schur functions
Use conjugacy to recover Van der Jeugt’s formula
Try to identify the origin of the row length restriction
ℓ(λ′ ) ≤ p in Macdonald’s formula
Try to identify the origin of the column length restriction
ℓ(λ) ≤ p in Van der Jeugt’s Conjecture
First some preliminaries on
Schur functions and Schur functions series
Partitions, Young diagrams, Frobenius notation
Determinantal identities and modifications
Nagoya - 2008
– p. 5/88
Schur functions
Let n be a fixed positive integer
Let x = (x1 , x2 , . . . , xn ) be a sequence of indeterminates
Let λ = (λ1 , λ2 , . . . , λn ) with λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0
be a partition of weight |λ| and length ℓ(λ) ≤ n
Then the Schur
function
sλ (x) is defined by:
λj +n−j xi
1≤i,j≤n
sλ (x) =
xn−j i
1≤i,j≤n
n−j Q
where xi 1≤i,j≤n = 1≤i<j≤n (xi − xj )
These Schur functions form a Z-basis of Λn , the ring of
polynomial symmetric functions of x1 , . . . , xn .
Nagoya - 2008
– p. 6/88
Partitions and Young diagrams
Young diagrams F λ consists of |λ| boxes arranged in
ℓ(λ) rows of lengths λi for i = 1, 2, . . . ℓ(λ)
Conjugate partition λ′ is the partition defined by the ℓ(λ′ )
columns of F λ of lengths λ′j for j = 1, 2 . . . , ℓ(λ′ )
Frobenius notation If F λ has r boxes on the main
diagonal, with arm and leg
lengths akand bk for
a1 a2 · · · ar
k = 1, 2, . . . , r, then λ =
has rank r(λ) = r
b1 b2 · · · br
with a1 > a2 > · · · > ar ≥ 0 and b1 > b2 > · · · > br ≥ 0
λ′1 λ′2 λ′3 λ′4 λ′5
λ1
=
λ2
λ3
=
λ4
a1
=
b1
a2
b2
a3
b3
Nagoya - 2008
– p. 7/88
Special families of partitions
Let P be the set of all partitions, including the zero
partition λ = 0 = (0, 0, . . . , 0).
The zero partition is the unique partition of weight, length
and rank zero, ie. |0| = ℓ(0) = r(0) = 0
Then for any integer t let


a1 a2 · · · ar
Pt = λ =
∈ P ak − bk = t

b1 b2 · · · br

for k = 1, 2, . . . , r 
and r = 0, 1, . . . 
Note: The zero partition belongs to Pt for all integer t
Nagoya - 2008
– p. 8/88
Modification rules
For n ∈ N let x = (x1 , x2 , . . . , xn ) and x = x1 x2 · · · xn
Let κ = (κ1 , κ2 , . . . , κn ) with κi ∈ Z for i = 1, 2, . . . , n
κj +n−j xi
1≤i,j≤n
Let sκ (x) =
xn−j i
1≤i,j≤n
Either sκ (x) = 0 or sκ (x) = ± xk sλ (x) for some
partition λ and some integer k
Permuting columns leads to various identities, such as
sκ (x) = −sµ (x) and sκ (x) = (−1)j−1 sν (x) with
µ = (κ1 , . . . , κj+1 −1, κj +1, . . . , κn )
ν = (κj+1 −j, κ1 +1, . . . , κj +1, κj+2 . . . , κn )
Nagoya - 2008
– p. 9/88
Example
If n = 4 and κ = (0, 4, 0, 9) then sκ (x) = (−1)3+1 sλ (x)
with λ = (6, 4, 2, 1) since
9 6 3
3 6
9
xi xi xi xi xi xi xi xi = 3 2
3 2
xi xi xi 1 xi xi xi 1 where just the ith row of each determinant has been
displayed
Alternatively, one can proceed iteratively using the
previous identities
s0409 (x) = − s6151 (x) = + s6421 (x)
Nagoya - 2008
– p. 10/88
Diagrammatically
Ex: sκ (x) = s0409 (x) = −s6151 (x) = +s6421 (x) = +sλ (x)
= −
= +
= +
= +
0 1 2 3 4 5
1 0 1 2
3 2 1 0 1 2 3 4 5
= +
1 0 1 2
2 1
with λ = (6421) =
52
31
3
Nagoya - 2008
– p. 11/88
Frobenius notation and modifications
Let κj = 0 unless j ∈ {b1 +1, b2 +1, . . . , br +1}
Let b1 > b2 > · · · > br ≥ 0 without loss of generality
Let κ(j) = ak + bk + 1 if j = bk +1 so that
κ = (0br , ar +br +1, 0br−1 −br −1 , . . . , a2+b2+1, 0b1 −b2 −1 , a1+b1+1)
Then, if a1 > a2 > · · · > ar ≥ 0,
sκ (x) = (−1)b1 +b2 +···+br sλ (x)
with
λ=
a1 a2 · · · ar
b1 b2 · · · br
and r = r(λ)
Nagoya - 2008
– p. 12/88
Example
For κ = (0, 4, 0, 9) we have κ = 0 unless j ∈ {2, 4}
Hence r = 2 and b1 = 3 > b2 = 1 ≥ 0
Since κ4 = a1 + b1 + 1 = 9 and κ2 = a2 + b2 + 1 = 4 we
have a1 = 5 > a2 = 2 ≥ 0
52
Hence if we set λ =
= (6, 4, 2, 1)
31
we have s0409 (x) = (−1)3+1 s6421 (x) as before
5
4
=
+
3
2
1
9
Nagoya - 2008
– p. 13/88
Schur function series
Littlewood [1940] For all n ≥ 1 and x = (x1 , x2 , . . . , xn ):
Y
Y
X
sλ (x) =
(1 − xi )−1
(1 − xj xk )−1
1≤i≤n
λ
1≤j<k≤n
(1 − xj xk )−1
λ even
Y
1≤j≤k≤n
X
Y
(1 − xj xk )−1
X
λ′ even
sλ (x) =
sλ (x) =
1≤j<k≤n
A partition is even if all its non-zero parts are even
The infinite sums over λ involve no restriction on
either ℓ(λ) or ℓ(λ′ ), but sλ (x) = 0 if ℓ(λ) > n.
Nagoya - 2008
– p. 14/88
Inverse Schur function series
Littlewood [1940] For all n ≥ 1 and x = (x1 , x2 , . . . , xn )
X
Y
Y
(−1)(|λ|+r(λ))/2 sλ (x) =
(1 − xi )
(1 − xj xk )
1≤i≤n
λ∈P0
X
(−1)
|λ|/2
sλ (x) =
λ∈P1
X
1≤j<k≤n
Y
(1 − xj xk )
Y
(1 − xj xk )
1≤j≤k≤n
(−1)
|λ|/2
sλ (x) =
1≤j<k≤n
λ∈P−1
These series are finite for all finite n
For finite n both ℓ(λ) and ℓ(λ′ ) are restricted,
since for λ ∈ Pt these differ by t
Nagoya - 2008
– p. 15/88
Determinantal identities
Littlewood [1940] For all n ≥ 1 and x = (x1 , x2 , . . . , xn )
n−j
n+j−1 x
X
− xi
i
n−j (−1)[|λ|+r(λ)]/2 sλ (x)
=
x i
λ∈P0
n−j
n+j
x
X
− xi i
n−j =
(−1)|λ|/2 sλ (x)
x i
λ∈P1
n−j
n+j−2
x
X
+ χ j>1 xi
i
n−j =
(−1)|λ|/2 sλ (x)
x i
λ∈P−1
the determinants are all n × n with i, j = 1, 2, . . . , n
1 if P is true
and, for any proposition P , χP =
0 if P is false
Nagoya - 2008
– p. 16/88
General determinantal identity
For all n ≥ 1 and x = (x1 , x2 , . . . , xn )
n−j
n+t+j−1
x
X
+ q χj>−t xi
i
n−j =
(−1)[|λ|−r(λ)(t+1)]/2 q r(λ) sλ (x)
x Lemma K [2008]
i
λ∈Pt
where t is any integer, and q is arbitrary
and the determinants are all n × n
so that i, j = 1, 2, . . . , n
The special cases:
q = −1, t = 0;
q = −1, t = 1;
q = 1, t = −1,
correspond to Littlewood’s previous formulae
Nagoya - 2008
– p. 17/88
Algebraic proof
n−j
n−j
2j−1+t+n−j
n+t+j−1
x
x
+ q χ j>−t xi
+ q χ j>−t xi
i
i
n−j n−j =
x x i
i
n X
X
X
r
(−1)(jr −1)+···+(j2 −1)+(j1 −1) q r sλ (x)
=
q sκ (x) =
r=0
κ
λ∈Pt
κj = 2j −1+t for j ∈ {j1 , j2 , . . . , jr } and κj = 0 otherwise
with n ≥ j1 > j2 > · · · > jr ≥ 1 − χt<0 t
j 1 − 1 + t j2 − 1 + t · · · j r − 1 + t
λ=
∈ Pt
j1 − 1
j2 − 1
···
jr − 1
r = r(λ)
|λ| = 2((j1 −1) + (j2 −1) + · · · + (jr −1)) + r(t + 1)
Nagoya - 2008
– p. 18/88
Example with n = 4 and t = 2
=
=
4−j
5+j
x + q χ
xi j>−2
i
4−j x i
3
xi − q x6i
x2i − q x7i
3
xi
x2i
xi − q x8i
xi 1 9 1 − q xi
s0000 − q (s3000 + s0500 + s0070 + s0009 )
+q 2 (s3500 + s3070 + s0570 + s3009 + s0509 + s0079 )
−q 3 (s3570 + s3509 + s3079 + s0579 ) + q 4 s3579
=
1 + q (−s3 + s41 − s511 + s6111 )
+q 2 (−s44 + s541 − s552 − s6411 + s6521 − s6622 )
+q 3 (s555 − s6551 + s6652 − s6663 ) + q 4 s6666
Nagoya - 2008
– p. 19/88
Example with n = 4 and t = 2 contd.
In Frobenius notation sλ (x) =
a1 a2 · · · ar
, we have
b1 b2 · · · br
2
3
4
5
1+q −
+
−
+
0
1
2
3
32
42
43
52
53
54
2
+q −
+
−
−
+
−
10
20
21
30
31
32
432
532
542
543
3
+q
−
+
−
210
310
320
321
5432
4
+q
3210
Nagoya - 2008
– p. 20/88
Example with n = 4 and t = −2
=
4−j
1+j
x + q χ
xi j>2
i
4−j x i
3
xi
x2i
xi − q x4i
3
xi
x2i
xi
5 1 − q xi
1 =
s0000 − q (s0030 + s0005 ) + q 2 s0035
=
1 + q (−s111 + s2111 ) − q 2 s2222
=
1−q
0
2
+q
Nagoya - 2008
1
3
−q
2
10
32
– p. 21/88
Row length restricted Schur function series
Theorem [Macdonald 79; Désarménien 87, Stembridge 90;
Bressoud 98, Okada 98]
For all n ≥ 1, x = (x1 , x2 , . . . , xn ) and p ≥ 0:
n−j
n+p+j−1
x
X
− xi
i
sλ (x) = n−j Q
Q
x
(1 − xi )
(1 − xj xk )
′
i
λ:ℓ(λ )≤p
1≤i≤n
1≤j<k≤n
n−j
n+2p+j
x
X
− xi
i
sλ (x) = n−j Q
x (1 − xj xk )
′
i
λ even :ℓ(λ )≤2p
X
sλ (x) =
λ′ even :ℓ(λ′ )≤p
1
2
1≤j≤k≤n
n−j
1 n−j
n+p+j−2
n+p+j−2
x
+ x
− xi
+ xi
i
i
2
n−j Q
x i
1≤j<k≤n (1 − xj xk )
Nagoya - 2008
– p. 22/88
Row length restricted Schur function series
Using the Lemma for given q and t as indicated, we find
Corollary
For all x = (x1 , x2 , . . .)
q = −1, t = p
P
[|µ|−r(µ)(p−1)]/2
X
(−1)
sµ (x)
µ∈Pp
Q
sλ (x) = Q
1≤i≤n (1 − xi )
1≤j<k≤n (1 − xj xk )
′
λ:ℓ(λ )≤p
q = −1, t = 2p + 1
P
[|µ|−r(µ)(2p)]/2
X
(−1)
sµ (x)
µ∈P2p+1
Q
sλ (x) =
1≤j≤k≤n (1 − xj xk )
′
λ even :ℓ(λ )≤2p
q = ±1, t = p − 1
P
[|µ|−r(µ)p]/2
X
(−1)
sµ (x)
µ∈Pp−1 :r(µ) even
Q
sλ (x) =
1≤j<k≤n (1 − xj xk )
′
′
λ even :ℓ(λ )≤p
Nagoya - 2008
– p. 23/88
Row length restricted Schur function series
Littlewood’s inverse Schur function series formulae then give:
Corollary For all x = (x1 , x2 , . . .)
q = −1, t = p
P
[|µ|−r(µ)(p−1)]/2
X
(−1)
sµ (x)
µ∈Pp
P
sλ (x) =
[|ν|+r(ν)]/2 s (x)
(−1)
ν
ν∈P
′
0
λ:ℓ(λ )≤p
q = −1, t = 2p + 1
P
[|µ|−r(µ)(2p)]/2
X
(−1)
sµ (x)
µ∈P2p+1
P
sλ (x) =
|ν|/2 s (x)
(−1)
ν
ν∈P1
′
λ even :ℓ(λ )≤2p
q = ±1, t = p − 1
P
[|µ|−r(µ)p]/2
X
(−1)
sµ (x)
µ∈Pp−1 :r(µ) even
P
sλ (x) =
|ν|/2 s (x)
(−1)
ν
ν∈P
′
′
−1
λ even :ℓ(λ )≤p
Nagoya - 2008
– p. 24/88
Column length restricted Schur function series
Using the involution ω : sλ (x) 7→ sλ′ (x) for all λ
and noting that λ ∈ Pt =⇒ λ′ ∈ P−t for all t, we have
Corollary
X
For all x = (x1 , x2 , . . .)
sλ (x) =
λ:ℓ(λ)≤p
X
sλ (x) =
λ′ even :ℓ(λ)≤2p
X
sλ (x) =
λ even :ℓ(λ)≤p
P
(−1)[|µ|−r(µ)(p−1)]/2 sµ (x)
µ∈P−p
P
P
ν∈P0
(−1)[|ν|+r(ν)]/2 sν (x)
(−1)[|µ|−r(µ)(2p)]/2 sµ (x)
µ∈P−2p−1
P
P
ν∈P−1
(−1)|ν|/2 sν (x)
µ∈P−p+1 :r(µ) even
P
ν∈P1
Nagoya - 2008
(−1)[|µ|−r(µ)p]/2 sµ (x)
(−1)|ν|/2 sν (x)
– p. 25/88
Column length restricted Schur function series
Littlewood’s inverse Schur function series formulae then give:
Corollary For all x = (x1 , x2 , . . .)
[|µ|−r(µ)(p−1)]/2
(−1)
sµ (x)
µ∈P−p
Q
sλ (x) = Q
(1 − xi ) 1≤j<k≤n (1 − xj xk )
1≤i≤n
λ:ℓ(λ)≤p
P
[|µ|−r(µ)(2p)]/2
X
(−1)
sµ (x)
µ∈P−2p−1
Q
sλ (x) =
1≤j<k≤n (1 − xj xk )
λ′ even :ℓ(λ)≤2p
P
[|µ|−r(µ)p]/2
X
(−1)
sµ (x)
µ∈P−p+1 :r(µ) even
Q
sλ (x) =
1≤j≤k≤n (1 − xj xk )
X
P
λ even :ℓ(λ)≤p
Note: The first of these was Van der Jeugt’s Conjecture
Nagoya - 2008
– p. 26/88
Column length restricted Schur function series
Using (q, t) = (−1, −p), (±1, −p + 1) and (−1, −2p − 1) in
our Lemma, we find
For all n ≥ 1, x = (x1 , x2 , . . . , xn ) and p ≥ 0:
Theorem
n−j
n−p+j−1
p
x
X
− (−1) χ j>p xi
i
sλ (x) = n−j Q
Q
x i
1≤i≤n (1 − xi )
1≤j<k≤n (1 − xj xk )
λ:ℓ(λ)≤p
X
sλ (x) =
λ even :ℓ(λ)≤p
1
2
n−j
1 n−j
n−p+j n−p+j x
− χ j≥p xi
+ 2 xi + χ j≥p xi
i
n−j Q
x i
1≤j≤k≤n (1 − xj xk )
n−j
n−2p+j−2
X
xi + χ j>2p+1 xi
sλ (x) = n−j Q
x ′
i
1≤j<k≤n (1 − xj xk )
λ even :ℓ(λ)≤2p
Nagoya - 2008
– p. 27/88
Row length restricted Schur function series
Alternative universal expressions giving each restricted series
as a product of an unrestricted series and a correction factor
for all x = (x1 , x2 , . . . ) take the form
X
sλ (x) =
λ:ℓ(λ′ )≤p
X
sλ (x) =
X
sλ (x) ·
λ
X
λ even :ℓ(λ′ )≤2p
λ even
X
X
sλ (x) =
λ′ even :ℓ(λ′ )≤p
X
(−1)[|µ|−r(µ)(p−1)]/2 sµ (x)
µ∈Pp
sλ (x) ·
X
(−1)[|µ|−r(µ)(2p)]/2 sµ (x)
µ∈P2p+1
sλ (x) ·
λ′ even
X
(−1)[|µ|−r(µ)p]/2 sµ (x)
µ∈Pp−1 :r(µ) even
Nagoya - 2008
– p. 28/88
Column length restricted Schur function series
Alternative universal expressions giving each restricted series
as a product of an unrestricted series and a correction factor
for all x = (x1 , x2 , . . . ) take the form
X
sλ (x) =
λ:ℓ(λ)≤p
X
sλ (x) =
λ′ even :ℓ(λ)≤2p
X
sλ (x) =
λ even :ℓ(λ)≤p
X
sλ (x) ·
sλ (x) ·
λ′ even
X
(−1)[|µ|−r(µ)(p−1)]/2 sµ (x)
µ∈P−p
λ
X
X
X
(−1)[|µ|−r(µ)(2p)]/2 sµ (x)
µ∈P−2p−1
sλ (x) ·
λ even
X
(−1)[|µ|−r(µ)p]/2 sµ (x)
µ∈P−p+1 :r(µ) even
Nagoya - 2008
– p. 29/88
Special case - row length ≤ 1
Setting p = 1 we recover a very well known series:
X
sλ (x) =
λ:ℓ(λ′ )≤1
=
Y
X
sλ (x) ·
−1
Y
1≤i≤n
=
Y
1≤i≤n
(−1)|µ|/2 sµ (x)
µ∈P1
λ
(1 − xi )
X
(1 − xj xk )
−1
1≤j<k≤n
(1 + xi ) =
n
X
s1m (x) =
m=0
·
Y
(1 − xj xk )
1≤j≤k≤n
n
X
em (x)
m=0
where the sums over m are finite, but could be extended
to ∞ since s1m (x) = em (x) = 0 for all m > n.
Nagoya - 2008
– p. 30/88
Special case - column length ≤ 1
Again setting p = 1 we recover another very well known
series:
X
sλ (x) =
λ:ℓ(λ)≤1
=
Y
1≤i≤n
=
Y
1≤i≤n
X
sλ (x) ·
(−1)|µ|/2 sµ (x)
µ∈P−1
λ
(1 − xi )
X
−1
Y
(1 − xj xk )
−1
1≤j<k≤n
(1 − xi )−1 =
∞
X
sm (x) =
m=0
·
Y
(1 − xj xk )
1≤j<k≤n
∞
X
hm (x)
m=0
where the sums over m are infinite since for all n ≥ 1 we
have sm (x) = hm (x) 6= 0 for any m ≥ 0.
Nagoya - 2008
– p. 31/88
So far
We have recast the numerator of Macdonald’s formula as
a signed sum of Schur functions
We have then used conjugacy to prove Van der Jeugt’s
conjecture
We have obtained three determinantal formulae on
column length restricted partitions analogous to those for
row length restricted partitions
We have not explained why the various determinants lead
to row or column length restrictions
To do this we need to exploit the fact that they define
characters of particular representations of classical
groups, as emphasized by Okada
Nagoya - 2008
– p. 32/88
Classical groups and their characters
Let x = (x1 , x2 , . . . , xn ) and x = (x1 , x2 , . . . , xn )
−ǫi
with xi = eǫi and xi = x−1
=
e
for i = 1, 2, . . . , n
i
λj +n−j xi
λ
n−j ch VGL(n) =
x i
1
1
λ
+n−j+
λ
+n−j+
j
j
2
2
− xi
xi
λ
=
ch VSO(2n+1)
1
1
n−j+ 2 n−j+ 2
x
−
x
i
i
λj +n−j+1
λj +n−j+1 x
−
x
i
i
λ
n−j+1
ch VSp(2n) =
n−j+1 x
− xi
i
λj +n−j
λ +n−j
λ +n−j λ +n−j + xi j
− xi j
xi
+ xi j
λ
n−j
ch VSO(2n)
=
n−j
x
+ xi i
Nagoya - 2008
– p. 33/88
Row length restricted series and characters
Theorem [Macdonald, Stembridge, Okada]
n−j
n+p+j−1
x
X
− xi
(p/2)n
i
p/2
sλ (x) = n−j
ch VSO(2n+1) (x, x, 1)
n+j−1 = x
xi − xi
λ:ℓ(λ′ )≤p
n−j
n+2p+j
x
X
− xi
i
pn
p
n−j
= x ch VSp(2n) (x, x)
sλ (x) =
n+j
x
−
x
′
i
i
λ even :ℓ(λ )≤2p
n−j
n−j
n+p+j−2
n+p+j−2
x
+ x
X
− xi
+ xi
i
i
n−j
sλ (x) =
n+j−2 x
+x
′
′
i
λ even :ℓ(λ )≤p
= x
where
p/2
(p/2)n
(−)n
ch VSO(2n)
i
(x, x)
n
1
x = x1 x2 · · · xn = ch VGL(n
(x)
Nagoya - 2008
– p. 34/88
Proof of formulae in terms of characters
Start from the original determinantal formulae
In each determinant permute columns under
j → n−j +1
Extract factors (−1)n by changing signs
of all terms of the form xai − xbi
Extract factors
n− 21 + p2
xi
xin+p
n−1+ p2
xi
and
n− 21
xi
and xni
and xin−1
from each row of numerator and denominator
determinants
Nagoya - 2008
– p. 35/88
Howe dual pairs of groups
Definition [Howe 85]
Let groups G and H act on a linear vector space V
Let their actions mutually commute
As a representation of G × H, let
λ(k)
V = ⊕k∈K VG
µ(k)
⊗ VH
k varies over some index set K
λ(k)
and VH
λ(k)
and VH
VG
VG
µ(k)
are irreps of G and H
µ(k)
vary without repetition
In such a case we say that G and H form a (Howe)
dual pair with respect to V .
Nagoya - 2008
– p. 36/88
Howe dual pairs of classical groups
In some cases V is an irrep of a group F ⊇ G × H
On restriction to the subgroup G × H
X
λ(k)
µ(k)
F
ch VG×H =
ch VG ch VH
k∈K
Ex: [Howe 89, Hasegawa 89] For V the spin irrep of an
othogonal group with character ch V ∆ , dual pairs are defined
through each of the following restrictions:
O(4np)
O(4np + 2p)
O(4np + 2n)
O(4np + 2n + 2p + 1)
O(4np)
⊇
⊇
⊇
⊇
⊇
Nagoya - 2008
SO(2n) × O(2p)
SO(2n + 1) × O(2p)
SO(2n) × O(2p + 1)
SO(2n + 1) × O(2p + 1)
Sp(2n) × Sp(2p)
– p. 37/88
n
Notation for p -complements
For any partition λ ⊆ np we have λ′ ⊆ pn
In such a case, let λ† = (p − λ′n , . . . , p − λ′2 , p − λ′1 )
Then λ† is also a partition
Ex: If p = 4, n = 5 and λ = (4, 3, 1)
then λ′ = (3, 2, 2, 1) and λ† = (4, 3, 2, 2, 1)
F
λ
=
F
λ†
∗
=
∗ ∗
∗ ∗
∗ ∗ ∗
Note: 0† = pn = (p, p, . . . , p)
Nagoya - 2008
– p. 38/88
The spin module and Howe dual pairs
Theorem [Morris 58,60; Hasegawa 89; Terada 93; Bump
and Gamburd 05] On restriction to the appropriate subgroup:
X
∆
λ†
λ
ch VO(4np) =
ch VSO(2n) ch VO(2p)
λ⊆np
∆
ch VO(4np+2p)
=
X
λ†
ch VSO(2n+1)
X
∆;λ†
ch VSO(2n)
X
∆;λ†
ch VSO(2n+1)
X
λ†
ch VSp(2n)
∆;λ
ch VO(2p)
λ⊆np
∆
ch VO(4np+2n)
=
λ
ch VO(2p+1)
λ⊆np
∆
ch VO(4np+2n+2p+1)
=
∆;λ
ch VO(2p+1)
λ⊆np
∆
ch VO(4np)
=
λ
ch VSp(2p)
λ⊆np
Nagoya - 2008
– p. 39/88
Exploitation of Howe duality
Let (G, H) be a Howe dual pair with F ⊇ G × H such
P
λ(k)
µ(k)
F
that ch VG×H = k∈K ch VG ch VH
λ(k)
µ(k)
The character ch VG
is just the coefficient of ch VH
F
in any formula we can devise for ch VG×H
In the case of the spin character identities all that is
needed are:
dual Cauchy formula
expressions for classical group characters in terms of
Schur functions [Littlewood 1940]
some modification rules [Newell 1951]
Nagoya - 2008
– p. 40/88
Spin characters and their decomposition
In terms of appropriate parameters
∆
ch VO(2n)
(x, x) =
n
Y
1
2
− 21
(xi + xi ) = x−1
i=1
n
Y
(1 + xi )
i=1
∆
ch VO(4np)
(xy, xy, xy, xy)
=
=
p
n Y
Y
i=1 j=1
p
n Y
Y
1
2
1
2
(xi yj +
1
− 12 − 12
− 12
2
xi yj )(xi yj
+
− 21 12
xi y j )
(xi + xi + yj + y j )
i=1 j=1
p
n Y
Y
−p
=x
(1 + xi yj )(1 + xi y j ) = x−p
i=1 j=1
X
sζ ′ (x) sζ (y, y)
ζ⊆n2p
Nagoya - 2008
– p. 41/88
Application to Howe dual pair contd.
=x
−p
X
sζ ′ (x) sζ (y, y) = x
−p
ζ⊆n2p
= x−p
X
= x−p
sζ ′ (x)
= x−p
η⊆n2p
=
X
X
ζ/β
ch VSp(2p) (y, y)
β:β ′ even
W2p
X
sη′ (x) sβ ′ (x)
β:β ′ even
η⊆n2p
X
ζ
sζ ′ (x) ch VGL(n)
(y, y)
ζ⊆n2p
ζ⊆n2p
X
X
W2p
X
sη′ (x) sδ (x)
δ even
λ†
ch VSp(2n) (x, x)
η
ch VSp(2p)
(y, y)
η
ch VSp(2p)
(y, y)
λ
ch VSp(2p)
(y, y) dual pair Theorem
λ⊆np
where W2p restricts any sum of Schur functions sν (x) to
those having ν1 = ℓ(ν ′ ) ≤ 2p
Nagoya - 2008
– p. 42/88
Character formula
It follows that
λ†
ch VSp(2n) (x, x)
= x−p
X
εη,λ W2p
η⊆n2p
X
sη′ (x) sδ (x)
δ even
where the modification rules for Sp(2p) characters are such
that
εη,λ =
±1
η
λ
if ch VSp(2p)
(y, y) = ± ch VSp(2p)
(y, y)
0
otherwise
In the special case λ = 0, so that λ† = pn this gives
X
X
pn
−p
−p
ch VSp(2n) (x, x) = x W2p
sδ (x) = x
δ even
Nagoya - 2008
sδ (x)
δ even:ℓ(δ ′ )≤2p
– p. 43/88
The spin module and Howe dual pairs
Thus we have recovered the Stembridge formula for the
symplectic group characters as a sum of row length
restricted Schur functions specified by even partitions
The formulae of Macdonald and Okada may be recovered
in the same way from the Howe dual pairs listed earlier
In each case the row length restriction owes its origin to
the bijective correspondence between irreps of the dual
groups specified by λ† and λ
We would like to identify other Howe dual pairs that might
lead to characters expressible as our sums of column
length restricted Schur functions
Such characters are necessarily infinite dimensional
Nagoya - 2008
– p. 44/88
The metaplectic module and Howe dual pairs
We need an infinite-dimensional analogue of the spin
representation of the orthogonal group
This is provided by the metaplectic representation of the
symplectic group
Ex: [Howe 89] For V the metaplectic irrep of a symplectic
˜
∆
group with character ch V , dual pairs are defined through
each of the following restrictions:
Sp(4np) ⊇ Sp(2n) × O(2p)
Sp(4np + 2p) ⊇ Sp(2n) × O(2p + 1)
Sp(4np) ⊇ SO(2n) × Sp(2p)
Nagoya - 2008
– p. 45/88
Metaplectic dual pair character formula
Theorem [Moshinsky and Quesne 71, Kashiwara and Vergne
78, Howe 85, K and Wybourne 85]
On restriction to the appropriate subgroup:
X
˜
p(λ)
∆
λ
ch VSp(2n) ch VO(2p)
ch VSp(4np) =
λ:λ′1 +λ′2 ≤2p, λ′1 ≤n
˜
∆
ch VSp(4np+2n)
=
X
p+ 21 (λ)
ch VSp(2n)
λ
ch VO(2p+1)
λ:λ′1 +λ′2 ≤2p+1, λ′1 ≤n
˜
∆
ch VSp(4np)
=
X
p(λ)
ch VSO(2n)
λ
ch VSp(2p)
λ:λ′1 ≤min(p,n)
Nagoya - 2008
– p. 46/88
Metaplectic characters and their decomposition
In terms of appropriate parameters
˜
∆
ch VSp(2n)
(x, x) =
n
Y
− 21
(xi
1
2
− xi )−1 = x
i=1
n
Y
(1 − xi )−1
i=1
˜
∆
ch VSp(4np) (xy, xy, xy, xy)
=
p
n Y
Y
− 12 − 12
(xi yj
i=1 j=1
p
n Y
Y
p
=x
1
2
1
2
− xi y j )
−1
− 21 21
(xi yj
−
1
2
− 12 −1
xi y j )
(1 − xi yj )−1 (1 − xi y j )−1
i=1 j=1
= xp
X
sζ (x) sζ (y, y)
ζ:ℓ(ζ)≤min(n,2p)
Nagoya - 2008
– p. 47/88
Application to Howe dual pair contd.
=x
p
X
sζ (x) sζ (y, y)
X
ζ
sζ (x) ch VGL(2p)
(y, y)
ζ:ℓ(ζ)≤min(n,2p)
=x
p
ζ:ℓ(ζ)≤min(n,2p)
= xp
X
X
sζ (x)
δ even
ζ:ℓ(ζ)≤min(n,2p)
= xp
X
L2p
η:ℓ(η)≤min(n,2p)
=
X
ζ/δ
ch VO(2p) (y, y)
X
δ even
sη (x) sδ (x)
η
ch VO(2p)
(y, y)
p(λ)
λ
ch VSp(2n) (x, x) ch VO(2p)
(y, y) dual pair
λ:λ′1 +λ′2 ≤2p, λ′1 ≤n
where L2p restricts any sum of Schur functions sν (x) to
those having ν1′ = ℓ(ν) ≤ 2p
Nagoya - 2008
– p. 48/88
Character formula
It follows that
p(λ)
ch VSp(2n) (x, x) = xp
X
η:ℓ(ζ)≤min(n,2p)
εη,λ L2p
X
sη (x) sδ (x)
δ even
where the modification rules for O(2p) characters are such
that
εη,λ =
±1
η
λ
if ch VO(2p)
(y, y) = ± ch VO(2p)
(y, y)
0
otherwise
In the special case λ = 0 this gives
X
p(0)
ch VSp(2n) (x, x) = xp L2p
sδ (x) = xp
δ even
Nagoya - 2008
X
sδ (x)
δ even:ℓ(δ)≤2p
– p. 49/88
The metaplectic module and Howe dual pairs
Thus we have obtained a formula for a particular
symplectic group character as a sum of column length
restricted Schur functions specified by even partitions
Our other column length restricted Schur function formula
may be also be identifed with characters in the same way
In each case the column length restriction owes its origin
to the bijective correspondence between irreps of the
dual groups specified by p(λ) and λ
Nagoya - 2008
– p. 50/88
Generalisation to Lie supergroups
Howe’s original work on dual pairs encompassed
supergroups, such as GL(m/n) and OSp(m/n)
All our identities can be extended to orthosymplectic
versions [Cheng and Zhang 04]
In particular, one can recover our starting point:
Proposition [Van der Jeugt, Lievens and Stoilova, 2007]
Let x = (x1 , x2 , . . . , xn ), then for each positive integer
p the irrep V (p) of OSp(1|2n) has character
X
p/2
sλ (x)
ch V (p) = (x1 x2 · · · xn )
λ:ℓ(λ)≤p
Nagoya - 2008
– p. 51/88
Cauchy formula and its inverse
Let m, n be positive integers
Then for all x = (x1 , x2 , . . . , xm ) and y = (y1 , y2 , . . . , yn )
X
sλ (x) sλ (y) =
(1 − xi yj )−1
m Y
n
Y
(1 − xi yj )
i=1 j=1
λ
X
m Y
n
Y
(−1)|λ| sλ (x) sλ′ (y) =
i=1 j=1
λ
The first sum over λ is infinite with non-zero terms
arising for all ℓ(λ) ≤ min{m, n}, and no restriction on ℓ(λ′ )
The second sum over λ is finite, with λ ⊆ nm , since
sλ (x) = 0 if ℓ(λ) > m and sλ′ (y) = 0 if ℓ(λ′ ) > n
Nagoya - 2008
– p. 52/88
Determinantal identity
For all x = (x1 , x2 , . . . , xm ) and y = (y1 , y2 , . . . , yn )
y j−1
n+1−i m Y
n
Y
1
m−j · · · · =
(1 − xi ya )
x
|yan−b | i
i=1 a=1
m+n−j
x
i−n
where the m+n × m+n determinant in the numerator is
partitioned after the nth row, and the determinants in the
denominator are m × m and n × n
Proof: Use the fact that the three determinants are
Vandermonde determinants in the m variables xi , the n
variables ya and the m+n variables xi and y a , with y a = ya−1
Nagoya - 2008
– p. 53/88
Nature of key determinant
1
1
1
1
x61
x62
x63
Ex: m = 3, n = 4, setting y a = ya−1
6 5
2
3
4
5
6 y4 y4 y4 y4 y4 y4 y 4 y 4
y3 y32 y33 y34 y35 y36 y 63 y 53
y2 y22 y23 y24 y25 y26 y 62 y 52
y1 y12 y13 y14 y15 y16 ∼ y 61 y 51
6 5
5
4
3
2
x1 x1 x1 x1 x1 1 x1 x1
6 5
5
4
3
2
x2 x2 x2 x2 x2 1 x2 x2
6 5
5
4
3
2
x3 x3 x3 x3 x3 1 x3 x3
for a = 1, 2, 3, 4
y 44 y 34 y 24 y 4 1
y 43 y 33 y 23 y 3 1
y 42 y 32 y 22 y 2 1
y 41 y 31 y 21 y 1 1
x41 x31 x21 x1 1
x42 x32 x22 x2 1
x43 x33 x23 x3 1
Factors (xi − xj ), (ya − yb ), (1 − xi ya ) ∼ (y a − xi )
Leading term y40 y31 y22 y13 x21 x12 x03
Nagoya - 2008
– p. 54/88
Corollary
For all x = (x1 , x2 , . . . , xm ) and y = (y1 , y2 , . . . , yn )
y j−1
n+1−i
X
1
|λ|
(−1) sλ (x) sλ′ (y) = m−j n−j · · · ·
x
y m
i
i
λ⊆n
xm+n−j
i−n
where the m+n × m+n determinant in the numerator is
partitioned after the nth row, and the determinants in the
denominator are m × m and n × n
Note: This may also be proved directly through using the
Laplace expansion of the determinant
Nagoya - 2008
– p. 55/88
Row length restricted Cauchy formula
Theorem Let x = (x1 , x2 , . . . , xm ) and y = (y1 , y2 , . . . , yn )
with m, n ≥ 1. Then for all p ≥ 0 we have
X
sλ (x) sλ (y)
λ:ℓ(λ′ )≤p
1
= m−j x
|yan−b | Qm Qn
i
i=1
a=1
y j−1+χ j>n p
n+1−i
· ···
(1 − xi ya ) m+n−j+χ p
j≤n
x
i−n
Since sλ (x) = 0 if ℓ(λ) > m and sλ (y) = 0 if ℓ(λ) > n
the sum is restricted to λ ⊆ pq with q = min{m, n}
If p = 0 then both sides are just 1, as confirmed by the
inverse Cauchy determinantal identity
Nagoya - 2008
– p. 56/88
Proof of row length restricted Cauchy formula
m+n−j+p ..
y n+1−i
.
n
Y
m+n−1+p =
ya
···
···
a=1
m+n−j+p ..
xi−n
.
m+n−j
y
n+1−i n
Y
m+n−1+p
=
ya
spn (y, x) · · · m+n−j a=1
x
i−n
y j−1
n+1−i n
Y
p
=
ya spn (y, x) · · · m+n−j a=1
x
i−n
y j−1+χ j>n p
n+1−i
···
m+n−j+χ j≤n p
x
i−n
Nagoya - 2008
m+n−j
y n+1−i
···
m+n−j
xi−n
– p. 57/88
Proof contd.
where
spn (y, x) =
X
sλ (x) spn /λ (y) =
λ⊆pn
so that
y j−1+χ j>n p
n+1−i
···
m+n−j+χ j≤n p
x
i−n
with
y j−1
n+1−i
···
m+n−j
x
i−n
n
Y
y pa
a=1
y j−1
n+1−i
···
m+n−j
x
i−n
=
X
X
λ⊆pn
sλ (x) sλ (y)
λ:ℓ(λ′ )≤p
n
m Y
m−j n−b Y
y a = x
(1 − xi ya )
i
i=1 a=1
Nagoya - 2008
sλ (x) sλ (y)
– p. 58/88
Extended dual Cauchy formula
Lemma [K, 2008]
Let x = (x1 , . . . , xm ) and y = (y1 , . . . , yn )
Then for each pair of integers p and q
j−1
y
n+1−i
1
m−j n−j · ···
x
y i
i
m+n−j+p
χ j≤n+p xi−n
=
X
we have
..
j−1+q
. χ j>n−q yn+1−i
···
..
.
m+n−j
xi−n
(−1)|ζ| sσ (x) sτ (y)
ζ⊆nm
where σ = (ζ + pr ) and τ = (ζ ′ + q r ) with r = r(ζ)
Nagoya - 2008
– p. 59/88
Extended dual Cauchy formula
where the determinant is (m + n) × (m + n), and is
partitioned after the nth row and nth column
a1 a2 · · · ar
and if ζ =
∈ (nm )
b1 b2 · · · br
with a1 < n, b1 < m and r
a1 +p a2 +p · · ·
σ=
b1
b2
···
b1 +q b2 +q · · ·
τ=
a1
a2 · · ·
= r(ζ), then
ar +p
br
br +q
ar
with ar ≥ max{0, −p} and br ≥ max{0, −q}
Nagoya - 2008
– p. 60/88
Examples of key determinant
m = 3, n = 4, p = 2, q = 1
1
y4
y42
y43
1
y3
y32
y33
1
y2
y22
y23
1
y1
y12
y13
···
···
···
···
x81
x71
x61
x51
x82
x72
x62
x52
x83
x73
x63
x53
..
.
..
.
..
.
..
.
..
.
..
.
..
.
y45
y46
y47
y35
y36
y37
y25
y26
y27
y15
y16
y17
···
···
···
x21
x1
1
x22
x2
1
x23
x3
1
Nagoya - 2008
– p. 61/88
Examples of key determinant
m = 3, n = 4, p = −2, q = 1
1
y4
y42
y43
1
y3
y32
y33
1
y2
y22
y23
1
y1
y12
y13
···
···
···
···
x41
x31
−
−
x42
x32
−
−
x43
x33
−
−
..
.
..
.
..
.
..
.
..
.
..
.
..
.
y45
y46
y47
y35
y36
y37
y25
y26
y27
y15
y16
y17
···
···
···
x21
x1
1
x22
x2
1
x23
x3
1
Nagoya - 2008
– p. 62/88
Examples of key determinant
m = 3, n = 4, p = 2, q = −1
1
y4
y42
y43
1
y3
y32
y33
1
y2
y22
y23
1
y1
y12
y13
···
···
···
···
x81
x71
x61
x51
x82
x72
x62
x52
x83
x73
x63
x53
..
.
..
.
..
.
..
.
..
.
..
.
..
.
−
y44
y45
−
y34
y35
−
y24
y25
−
y14
y15
···
···
···
x21
x1
1
x22
x2
1
x23
x3
1
Nagoya - 2008
– p. 63/88
Examples of key determinant
m = 3, n = 4, p = −2, q = −1
1
y4
y42
y43
1
y3
y32
y33
1
y2
y22
y23
1
y1
y12
y13
···
···
···
···
x41
x31
−
−
x42
x32
−
−
x43
x33
−
−
..
.
..
.
..
.
..
.
..
.
..
.
..
.
−
y44
y45
−
y34
y35
−
y24
y25
−
y14
y15
···
···
···
x21
x1
1
x22
x2
1
x23
x3
1
Nagoya - 2008
– p. 64/88
Ex 1: m = 3, n = 4, p = 2, q = 1
Ex. π =
1 y4 y42
1 y3 y32
1 y2 y22
1 y1 y12
·
·
·
1 2 3 4 5 6 7
2 3
3 ..
y4 .
3 ..
y3 .
3 ..
y2 .
3 ..
y .
1
·
..
.
8
7
6
5 ..
x2 x2 x2 x2 .
8
7
6
5 ..
x3 x3 x3 x3 .
x81
x71
x61
x51
5 7 1 4 6
y45 y46 y47 y y2 y5 y7 y35 y36 y37 4 4 4 4 8 5
x1 x1 x1 y y2 y5 y7 y25 y26 y27 3 3 3 3 8 5
∼ − · x2 x2 x2 y2 y 2 y 5 y 7 y15 y16 y17 2
2
2 5
8
x x x3 3
3
2
5
7 y1 y1 y1 y1
· · · 2
x1 x1 1 4−b 3−j x22 x2 1 = − s4311 (y) ya · s641 (x) xi 2
x3 x3 1 Nagoya - 2008
– p. 65/88
Ex 1: m = 3, n = 4, p = 2, q = 1
Ex. π =
1 2 3 4 5 6 7
(−1)π = (−1)|ζ| = −1
2 3 5 7 1 4 6
3 0
ζ = (4, 2, 1) =
2 0
∗ ∗
3+2 0 + 2
5 2
σ=
=
= (6, 4, 1)
∗ ∗
2
0
2 0
2 0
′
ζ = (3, 2, 1, 1) =
3 0
∗
∗
τ=
2+1 0+1
3
0
Nagoya - 2008
=
3 1
3 0
= (4, 3, 1, 1)
– p. 66/88
Ex 2: m = 3, n = 4, p = −2, q = −1
Ex. π =
1 y4 y42
1 y3 y32
1 y2 y22
1 y1 y12
·
·
x41
x31
x42 x32
x43 x33
·
1 2 3 4 5 6 7
3 4 6
3 ..
y4 . −
3 ..
y3 . −
3 ..
y2 . −
3 ..
y . −
1
·
·
.. 2
− − . x1
.. 2
− − . x2
.. 2
− − . x3
7 1 2 5
y44 y45 y 2 y3
y34 y35 4 4
4
5 y2 y3
y2 y2 ∼ − 3 3
y2 y3
y14 y15 2 2
2 3
y1 y1
· · x1 1 x2 1 = − s2222 (y)
x3 1 Nagoya - 2008
y44 y45
y34 y35
y24 y25
y14 y15
4 3 2 x1 x1 x1 4 3 2 · x2 x2 x2 4 3 2 x3 x3 x3
4−b 3−j ya · s222 (x) x i
– p. 67/88
Ex 2: m = 3, n = 4, p = −2, q = −1
Ex. π =
1 2 3 4 5 6 7
(−1)π = (−1)|ζ| = +1
3 4 6 7 1 2 5
3 2
ζ = (4, 4, 2) =
2 1
∗ ∗
3−2 2−2
1 0
σ=
=
= (2, 2, 2)
∗ ∗
2
1
2 1
2 1
′
ζ = (3, 3, 2, 2) =
3 2
∗
∗
τ=
2−1 1−1
3
2
=
Nagoya - 2008
1 0
3 2
= (2, 2, 2, 2)
– p. 68/88
Row length restricted Cauchy formula
We can combine our Theorem for p ≥ 0 with our Lemma
for p = q ≥ 0
X
sλ (x) sλ (y)
λ:ℓ(λ′ )≤p
y j−1+χ j>n p
n+1−i
1
=
·
Q
Q
···
n
xm−j |yan−b | m
m+n−j+χ j≤n p
i
i=1
a=1 (1 − xi ya )
x
i−n
X
ζ
(−1)|ζ| sζ+pr (x) sζ ′ +qr (y)
j−1
y
n+1−i
1
= m−j · ···
x
|yan−b | i
χ j≤n+p xm+n−j+p
i−n
Nagoya - 2008
..
j−1+q
. χ j>n−q yn+1−i
···
..
.
m+n−j
xi−n
– p. 69/88
Row length restricted Cauchy formula
For all x = (x1 , x2 , . . . ), y = (y1 , y2 , . . . ) and p ≥ 0
X
sλ (x) sλ (y)
λ:ℓ(λ′ )≤p
= Qm Qn
i=1
=
X
λ
1
a=1
·
(1 − xi ya )
sλ (x) sλ (y) ·
X
X
(−1)|ζ| sζ+pr (x) sζ ′ +pr (y)
ζ
(−1)|ζ| sζ+pr (x) sζ ′ +pr (y)
ζ
This expresses the row length restricted series as a
product of the unrestricted series times a correction factor
Nagoya - 2008
– p. 70/88
Column length restricted Cauchy formula
Using the involutions ωx : sλ (x) 7→ sλ′ (x) and
ωy : sλ (y) 7→ sλ′ (y) for all λ, either separately or
together, we obtain three more restricted formula.
Using ωx ωy we find that for all x, y and for all p ≥ 0
X
sλ (x) sλ (y)
λ:ℓ(λ)≤p
=
X
X
sλ (x) sλ (y) ·
(−1)|ζ| s(ζ+pr )′ (x) s(ζ ′ +pr )′ (y)
X
X
sλ (x) sλ (y) ·
(−1)|η| sη−pr (x) sη′ −pr (y)
λ
=
λ
ζ
η
where the sum over η is restricted to those η such that
both η − pr and η ′ − pr are partitions
Nagoya - 2008
– p. 71/88
Column length restricted Cauchy formula
Using our Lemma with both p and q set equal to −p, with
p ≥ 0 gives
Theorem Let x = (x1 , x2 , . . . , xm ) and y = (y1 , y2 , . . . , yn )
with m, n ≥ 1. Then for all p ≥ 0 we have
1
X
sλ (x) sλ (y) = m−j x
|yan−b | Qm Qn (1 − xi ya )
i
i=1
a=1
λ:ℓ(λ)≤p
j−1
y
n+1−i
· ···
m+n−j−p
χ j≤n−p xi−n
Nagoya - 2008
..
j−1−p
. χ j>n+p yn+1−i
···
..
.
m+n−j
xi−n
– p. 72/88
Qutients by infinite series
So far we have discussed infinite series of Schur
functions, including their expression in determinantal form
These have been used to provide generalisations of
formulae of both Littlewood and Cauchy
Characters of the orthogonal and symplectic groups may
be expressed as quotients of given Schur functions by
infinite series of Schur functions in Pt for t = 0, ±1
Each of these characters has a simple determinantal
form, given by Weyl’s character formula
It is natural to ask if the same is true if t is extended to
other positive and negative integer values.
Nagoya - 2008
– p. 73/88
Bressoud-Wei identities
Bressoud and Wei [1992] For all integers t ≥ −1:
(t−|t|)/2 (t+|t|)/2
2
hλi −i+j (x) + (−1)
hλi −i−j+1−t (x) =
X
(−1)[|σ|+r(σ)(|t|−1)]/2 sλ/σ (x)
σ∈Pt
Hamel and K [2008] For all integers t and all q:
| hλi −i+j (x) + q χj>−t hλi −i−j+1−t (x) |
=
X
(−1)[|σ|−r(σ)(t+1)]/2 q r(σ) sλ/σ (x)
σ∈Pt
Nagoya - 2008
– p. 74/88
Algebraic proof
| hλi −i+j (x) + q χj>−t hλi −i−j+1−t (x) |
n X
X
r =
q hλi −i+j−κj (x) r=0
=
X
κ
(−1)
(jr −1)+···+(j2 −1)+(j1 −1)
σ∈Pt
r
q hλi −i+j−σj (x) κj = 2j−1+t for j ∈ {j1 , j2 , . . . , jr } and κj = 0 otherwise
with n ≥ j1 > j2 > · · · > jr ≥ 1 − χt<0 t
j 1 − 1 + t j2 − 1 + t · · · j r − 1 + t
σ=
∈ Pt
j1 − 1
j2 − 1
···
jr − 1
r = r(σ)
Nagoya - 2008
– p. 75/88
Cauchy-type extension
Theorem [Hamel and K, 2008]
Let x = (x1 , . . . , xm ) and y = (y1 , . . . , yn )
Let λ and µ have lengths ℓ(λ) ≤ m and ℓ(µ) ≤ n
Then for each pair of integers p and q we have
..
hµn+1−i +i−j (y)
. χ j>n−q hµn+1−i +i−j−q (y)
···
···
..
χ j≤n+p hλi−n −i+j−p (x) .
hλi−n −i+j (x)
=
X
(−1)|ζ| sλ/σ(ζ) (x) sµ/τ (ζ) (y)
ζ⊆nm
Nagoya - 2008
– p. 76/88
Cauchy-type extension
where the determinant is (m + n) × (m + n), and is
partitioned after the nth row and nth column
a1 a2 · · · ar
and if ζ =
∈ (nm )
b1 b2 · · · br
with a1 < n, b1 < m and r = r(ζ), then
a1 +p a2 +p · · · ar +p
r
σ(ζ) = ζ + p =
b1
b2
···
br
b1 +q b2 +q · · · br +q
′
r
τ (ζ) = ζ + q =
a1
a2 · · ·
ar
with ar ≥ max{0, −p} and br ≥ max{0, −q}
Nagoya - 2008
– p. 77/88
Example
m = 3, n = 4, p = −2, q = −1 λ = (5, 3, 2), µ = (4, 3, 2, 2)
Let {k} = hk (x) and {k} = hk (y) for all integers k
{7}
..
{1} {0} − .
..
{2} {1} {0} .
..
{4} {3} {2} .
..
{6} {5} {4} .
···
···
{2}
{3}
{5}
···
···
{3} {4}
−
−
{0} {1}
−
−
−
−
−
−
−
−
−
−
−
−
−
{1} {0}
−
{3} {2}
···
···
···
..
. {5} {6} {7}
..
. {2} {3} {4}
..
. {0} {1} {2}
Nagoya - 2008
– p. 78/88
Typical term in Laplace expansion
π=
{2}
{3}
{5}
{7}
1 2 3 4 5 6 7
(−1)π = (−1)0+1+1+2 = +1
1 3 4 6 2 5 7
{0} −
− {4} {5} {7}
{1} {0} − × {1} {2} {4}
{3} {2} {1} − {0} {2}
{5} {4} {3} (ζn′ , . . . , ζ1′ | ζ1 , . . . , ζm ) =
(1−1, 3 −2, 4 −3, 6 −4 | 5 −2, 6 −5, 7 −7) = (0, 1, 1, 2 | 3, 1, 0)
5−2−1
2
ζ = (3, 1, 0) =
=
= (−1)|ζ| = +1
6−4−1
1
r(ζ) = 1
Nagoya - 2008
– p. 79/88
Determination of σ(ζ) and τ (ζ)
ζ = (310) =
2
1
∗ ∗
r
=⇒ σ(ζ) = ζ + p = (310) − (200) = (110) =
ζ ′ = (2110) =
′
1
2
0
1
∗
r
=⇒ τ (ζ) = ζ + q = (2110) − (1000) = (1110) =
Nagoya - 2008
0
2
– p. 80/88
Identification of constituent determinants
Recall that λ = (532) and µ = (4322)
while σ(ζ) = (110) and τ (ζ) = (1110)
{4} {5} {7} ∗
∗
{1} {2} {4} = s532/110 (x)
− {0} {2} {2} {0} −
− {3} {1} {0} − = s4322/1110 (y)
{5} {3} {2} {1} {7} {5} {4} {3} Nagoya - 2008
∗
∗
∗
– p. 81/88
Conclusions
Both the classical Schur function series of Littlewood and
the Cauchy identity may be restricted with respect to row
lengths or column lengths through determinantal formulae
In each case the correction factors to the original
multiplicative formulae may be expressed as a signed
sum of Schur functions or pairs of Schur functions
specified by partitions having a particularly simple form in
Frobenius notation
Each row or column restricted Schur function series is
nothing other than the character of some (rather simple)
finite or infinite-dimensional irrep of a classical group
Nagoya - 2008
– p. 82/88
Conclusions
To evaluate these characters (and thereby derive the
restricted Schur function series) use may be made of
Howe dual pairs with respect to spin and metaplectic
representations of (the covering groups) of the orthogonal
and symplectic groups
All the Schur function identities may be extended to the
case of supersymmetric Schur functions
At present a dual pair approach has not been given for
the row or column restricted Cauchy identities
The dual pair approach enables many other characters to
be evaluated, although in doing so it is usually necessary
to invoke classical group modification rules
Nagoya - 2008
– p. 83/88
Some references on background and motivation
D.E. Littlewood, The Theory of Group Characters, Clarendon Press,
Oxford, 1940
A.J. Bracken and H.S. Green Algebraic identities for parafermi
statistics of given order Nuovo Cimento 9A (1972) 349-365
N.I. Stoilova and J. Van der Jeugt, The parafermion Fock space and
explicit so(2n+1) representations, J. Phys. A 41 (2008) 075202 (13
pp).
S. Lievens, N.I. Stoilova and J. Van der Jeugt, The paraboson
Fock space and unitary irreducible representations of the Lie superalgebra
osp(1—2n)
Commun. Math. Phys. 281 (2008) 805-826.
D.M. Bressoud and S.Y. Wei, Determinental formulae for complete
symmetric functions J. Comb. Theory A. 60 (1992) 277-286.
Nagoya - 2008
– p. 84/88
Row length restricted series
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford
University Press, Oxford, 1979
J. Désarménien, Une generalisation des formules de Gordon et de
MacMahon, C.R. Acad. Sci. Paris Ser.1 309 (1989) 269-272
J.R. Stembridge, Hall-Littlewood functions, plane partitions, and the
Rogers-Ramanujan identoties, Trans. Amer. Math. Soc. 319 (1990)
469-498
S. Okada, Applications of minor summation formulas to
rectangular-shaped representations of classical groups, J. Algebra 205
(1998) 337-367
D.M. Bressoud, Elementary proofs of identities for Schur functions and
plane partitions, Ramanujan J. 4 (2000) 69–80
Nagoya - 2008
– p. 85/88
Complementary groups or dual pairs
M. Moshinsky and C. Quesne, Noninvariance groups in the
second-quantisation picture and their applications, J. Math. Phys. 11
(1970) 1631-1639
M. Moshinsky and C. Quesne, Linear canonical transformations and
their unitary representations, J. Math. Phys. 12 (1971) 1772-1780
R. Howe, Dual pairs in Physics, Lect. in Applied Math. 21 Amer.
Math. Soc. (1985) 179-207
R. Howe, Transcending classical invariant theory, J. Amer. Math.
Soc. 2 (1989) 535-552
Nagoya - 2008
– p. 86/88
Dual pairs in spin modules
A. O. Morris, Spin representations of a direct sum and a direct product,
J. Lond. Math. Soc. 33 (1958) 326-333
A. O. Morris, Spin representations of a direct sum and a direct product
(II), Quart. J. Math. Oxford (2) 12 (1961) 169-176
K. Hasegawa, Spin module versions of Weyl’s reciprocity theorem for
classical Kac-Moody Lie algebras - an application to branching rule duality
Publ. RIMS, Kyoto Univ 25 (1989) 741-828
I. Terada, A Robinson-Schensted-type correspondence for a dual pair on
spinors, J. Combinatorial Theory 63 (1991) 90-109
Nagoya - 2008
– p. 87/88
Dual pairs in metaplectic modules
M. Kashiwara and M. Vergne, On the Segal-Shale-Weil
representations and harmonic polynomials, Inventiones Math. 31
(1978) 1-48
D. J. Rowe, B. G. Wybourne and P. H. Butler, Unitary
representations, branching rules and matrix elements for the non-compact
symplectic groups,
J. Phys. A 18 (1985) 939-953
R. C. King and B. G. Wybourne, Holomorphic discrete series and
harmonic series unitary irreducible representations of non-compact Lie
Sp(2n, R), U (p, q) and SO ∗ (2n), J. Phys. A 18 (1985)
3113-3139
groups:
Nagoya - 2008
– p. 88/88