Download Hall-Littlewood positive homomorphisms of the algebra of symmetric

Hall-Littlewood positive homomorphisms
of the algebra of symmetric functions
Alexey Bufetov
Department of Mathematics, Higher School of Economics, Moscow
May 25, 2015
Algebra of symmetric functions
{xi }∞
i=1 — formal variables.
Newton power sums:
pk :=
∞
X
xik .
i=1
The algebra of symmetric functions Λ := R[p1 , p2 , . . . ].
We shall consider homomorphisms ( = multiplicative
functionals ) π : Λ → R.
Examples.
P
1) x1 , x2 , . . . are nonnegative numbers such that i xi ≤ 1.
2) π(p1 ) = 1, π(pk ) = 0 for k ≥ 2.
Homomorphisms are in one to one correspondence with a
sequence {π(pk )}k=1,2,... .
Hall-Littlewood and Schur functions
Let λ = λ1 ≥ λ2 ≥ · · · ≥ λk be a partition (= Young
diagram). For t ∈ [0; 1) a Hall-Littlewood function is
determined by
Qλ (x1 , . . . , xn ; t) := cλ,t
X
λk
λ1
λ2
xσ(1)
xσ(2)
. . . xσ(k)
σ∈Sn
Y xσ(i) − txσ(j)
i<j
xσ(i) − xσ(j)
For t = 0 Hall-Littlewood functions turn into Schur functions:
λj +k−j n
det xi
i,j=1
Qλ (x1 , . . . , xn ; 0) = sλ (x1 , . . . , xn ) := Q
(x
−
xj )
1≤i<j≤n i
One can define Qλ ∈ Λ, sλ ∈ Λ.
.
Hall-Littlewood positive
homomorphisms
We shall say that π : Λ → R is a Hall-Littlewood positive
homomorphism if for any partition λ the value π(Qλ ) is
nonnegative.
We shall say that π : Λ → R is a Schur positive
homomorphism if for any partition λ the value π(sλ ) is
nonnegative.
HL-positive homomorphisms.
P
1) x1 , x2 , . . . are nonnegative numbers such that i xi ≤ 1.
2) π(p1 ) = 1, π(pk ) = 0 for k ≥ 2.
Classification
We are interested in the description of all Hall-Littlewood
positive homomorphisms.
This question arises in different contexts:
Schur case: representation theory of the infinite
symmetric group.
Schur case: totally positive Toeplitz matrices.
Hall-Littlewood case: asymptotic representation theory of
the general linear groups over a finite field.
Also Schur (or Macdonald) positive homomorphisms play an
important role in the construction of Schur (or Macdonald)
processes.
Schur case: Thoma theorem
Theorem (Thoma ’64, Edrei’53, Vershik-Kerov’81)
All Schur-positive homomorphisms of Λ are parameterized by
γ ≥ 0, and α = (α1 ≥ α2 ≥ . . . ≥ 0),P
β = (β1 ≥ β2 ≥ . . . ≥ 0), such that ∞
i=1 (αi + βi ) < ∞, and
∞
X
πα,β (p1 ) =
(αi + βi ) + γ,
i=1
πα,β (pk ) =
∞
X
i=1
αik
k−1
+ (−1)
∞
X
i=1
βik ,
k ≥2
Kerov’s conjecture
Conjecture (Kerov’93)
For 0 ≤ t < 1 all Hall-Littlewood positive homomorhpisms are
parameterized by γ ≥ 0, and α = (αP
1 ≥ α2 ≥ . . . ≥ 0),
β = (β1 ≥ β2 ≥ . . . ≥ 0), such that ∞
i=1 (αi + βi ) < ∞, and
!
∞
∞
X
X
1
πα,β (p1 ) =
,
αi + γ +
βi
1
−
t
i=1
i=1
πα,β (pk ) =
∞
X
i=1
αik +
∞
(−1)k−1 X k
β ,
1 − t k i=1 i
k ≥ 2.
Probability measures
Yn — the set of all Young diagrams with n boxes.
Let w = {αi , βj , γ} denote our set of P
parameters. Without
βi
γ
loss of generalization we assume that i (αi + 1−t
) + 1−t
= 1.
wPl corresponds to γ = 1 − t; αi = 0, βj = 0.
Let πw be a HL-positive homomorphism. Let us define a
probability measure on Yn by
HLwn (λ) := n!πwPl (Pλ )πw (Qλ ),
where Pλ is a “P”-Hall-Littlewood function (a constant
multiple of Qλ ).
It turns out that the probabilistic properties of these measures
are closely related to Kerov’s conjecture.
Law of Large Numbers: Schur case
Schur case:t = 0.
Parameters w = {αi , βj , γ} have the following probabilistic
meaning
Theorem (Vershik-Kerov’81)
Let λ(n) be the random Young diagram with n boxes
distributed according to HLwn . As n → ∞, we have
λi (n)
→ αi ,
n
λ0j (n)
→ βj ,
n
convergence in probability.
where λi (n) is the length of the i-th row of λ, and λ0j (n) is the
length of the j-th column of λ.
LLN: Hall-Littlewood case
w = {αi , βj , γ}. Let λ be a random Young diagram
distributed according to HLwn , and let γ = 0.
Theorem (Bufetov-Petrov’14)
As n → ∞, we have
λi (n)
→ αi ,
n
λ0j (n)
βj
→
,
n
1−t
convergence in probability.
Method of proof: — new dynamics (generalization of RSK)
which samples HLwn ; it is constructed with the use of
Borodin-Petrov’13 (similar dynamics were constructed in
O’Connell-Pei’12; related to q-Whittaker functions).
— Analysis of this dynamics.
(r, c)
λ
(i, j)
(r, c)
λ̂
(î, ĵ)
µ
(i, j)
µ̂
(î, ĵ)
Figure: An example of λ, λ̂ and µ, µ̂. Here the gray box is
ˆ j)
ˆ = (4, 2).
(r , c) = (1, 6), and (i, j) = (2, 4), (i,
1N stands for (1, 1, . . . , 1). Assume that N 1.
|
{z
}
N
Conjecture (Bufetov-Gorin’14)
Let λ, λ̂ ∈ Yn and µ, µ̂ ∈ Yn−1 be two pairs of Young diagrams
such that both λ,λ̂ and µ,µ̂ differ by the move of box (i, j)
ˆ j)
ˆ with iˆ > i. Further, assume that
into the position (i,
λ \ µ = λ̂ \ µ̂ = (r , c). If r < i, then
1−t
λ̂0c −λ̂0c+1
Q (1N ; t) N
µ̂
λ0c −λ0c+1 Qµ (1 ; t)
≥
1
−
t
.
Qλ̂ (1N ; t)
Qλ (1N ; t)
Proposition (Bufetov-Gorin’14)
The conjecture above implies Kerov’s conjecture.
Schur case
Proposition (Bufetov-Gorin’14)
Let λ, λ̂ ∈ Yn and µ, µ̂ ∈ Yn−1 be two pairs of Young
diagrams, such that both λ,λ̂ and µ,µ̂ differ by the move of
ˆ j)
ˆ with iˆ > i. Further, assume
box (i, j) into the position (i,
that λ \ µ = λ̂ \ µ̂ = (r , c). If r < i, then
sλ (1N )sµ̂ (1N ) ≥ sλ̂ (1N )sµ (1N ).
This proposition and considerations of Bufetov-Gorin’14 give a
new proof of Thoma theorem.
More information
More details can be found in
A. Bufetov, L. Petrov, “Law of Large Numbers for Infinite
Random Matrices over a Finite Field”, arXiv:1402.1772,
to appear in Selecta Mathematica.
A. Bufetov, V. Gorin, “Stochastic monotonicity in Young
graph and Thoma theorem”, arXiv:1411.3307, to appear
in IMRN.