LLT polynomials in a nutshell: on Schur expansion of LLT...
Transcript of LLT polynomials in a nutshell: on Schur expansion of LLT...
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LLT polynomials in a nutshell:
on Schur expansion of LLT polynomials
Meesue Yoo (Chungbuk National University)
July 14, 2020
@ KIAS Workshop on Combinatorial Problems of Algebraic Origin
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The goal
We first define the LLT polynomials,
explain why they are important by showing some examples
and, keep track of what have been known and what’s still open.
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Table of contents
1. Definition of LLT polynomials
2. Several important subfamilies of LLT polynomials
3. Schur positivity of LLT polynomials
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Definition of LLT polynomials
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Basic definitions
• A partition is a sequence of positive integers λ = (λ1, λ2, . . . , λn)
such that
λ1 ≥ λ2 ≥ · · · ≥ λn > 0.
• We identify a partition λ with its Young diagram.
Example. λ = (5, 4, 4, 2)
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More definitions
• Given two partitions λ and µ such that µ ⊆ λ, the skew shape
λ/µ is defined to be the set theoretic difference λ \ µ.
Example.
(5, 4, 4, 2)/(4, 2, 1)
• A semistandard Young tableau of shape ν = λ/µ is a function
T : ν→ Z>0 such that
∨≤
• SSYT(ν) = the set of semistandard Young tableaux of shape ν
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Symmetric functions
• A symmetric function is a polynomial f (x1, . . . , xn) which is
invariant under the action of the symmetric group
f (xσ(1), . . . , xσ(n)) = f (x1, . . . , xn),
or, σf = f for all σ ∈ Sn.
• The elementary symmetric functions eλ are defined by
en = ∑i1<···<in
xi1 · · · xin , n ≥ 1,
eλ = eλ1eλ2· · · , for λ = (λ1, λ2, . . . ).
• The Schur function of shape λ/µ is defined by
sλ/µ(x) = ∑T∈SSYT(λ/µ)
xT
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LLT polynomials
• LLT polynomials are a family of symmetric functions introduced by
Lascoux, Leclerc and Thibon in ’1997.
• The original definition uses cospin statistic of ribbon tableaux.
12
35
123
3
56
2 2
6
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LLT polynomials
• A ribbon tableau maps to a tuple of skew tableaux via quotient
map.
54321
0 -1 -2 -3 -4 -5 -6 -7
12
35
123
3
56
4 5
(1 0
-1, , )2 3
5 51 1 53 3
2 46
• Here, we use the variant definition of Bylund and Haiman using
inversion statistic.
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• Consider a k-tuple of skew diagrams ν = (ν(1), . . . , ν(k)), and let
SSYT(ν) = SSYT(ν(1))× · · · × SSYT(ν(k)).
• The content of a cell u = (row, column) = (i , j) is the integer
c(u) = i − j .
• Given T ∈ (T (1), . . . ,T (k)) ∈ SSYT(ν), a pair of entries
T (i)(u) > T (j)(v) form an inversion if either
(i) i < j and c(u) = c(v), oru
v
(ii) i > j and c(u) = c(v) + 1.v
u
• inv(T ) = the number of inversions in T .
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Definition of LLT polynomials
Definition
The LLT polynomial indexed by ν = (ν(1), . . . , ν(k)) is
LLTν(x ; q) = ∑T∈SSYT(ν)
qinv(T )xT .
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Note on properties of LLT polynomials
• LLT polynomials are symmetric.
• When q = 1, LLTν(x ; 1) = ∏ki=1 sv (i)(x).
• Furthermore, they are Schur positive. [Grojnowski-Haiman]
• However, combinatorial description for Schur coefficients are not
known.
• Macdonald polynomials Hµ(x ; q, t) can be expanded positively in
terms of LLT polynomials.
• Hence, the Schur expansion of LLT polynomial gives a combinatorial
formula for q, t-Kostka polynomials.
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Several important subfamilies of
LLT polynomials
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We consider LLT polynomials indexed by tuple of certain shapes:
ribbons, vertical strips and unicellular diagrams
⊃ ⊃
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1 LLT polynomials indexed by ribbons
⇐⇒1
2
3
4
5
6
7
8
9
10
11
≤
≤≤≤≤
12
34
56
78
910
11
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1 LLT polynomials indexed by ribbons
To read the inversion statistic:
⇐⇒1
5
3
4
6
2
4
8
7
6
2
≤
≤≤≤≤
15
34
62
48
76
2
q
q qq
qq q
q q
q9x1x22 x3x
24 x5x
26 x7x8
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Macdonald polynomials can be expanded positively in terms of
ribbon LLT’s
Hµ(X ; q, t): the modified Macdonald polynomial
Theorem [Haglund-Haiman-Loehr, ’2005]
Hµ(X ; q, t) = ∑σ: µ→Z+
qinv(σ)tmaj(σ)xσ
= ∑D⊆{(i ,j)∈µ: i>1}
q−a(D)tmaj(σ) LLTν(µ,D)(X ; q),
where ν(µ,D) = (ν(1), . . . , ν(k)), a tuple of ribbons.
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A tuple of ribbons ν(µ,D) = (ν(1), . . . , ν(k))
• Define the descent set of a ribbon ν to be the set of contents c(u)
of those cells u = (i , j) ∈ ν such that the cell v = (i − 1, j) directly
below u also belongs to ν.
Example.
ν =
,
contents=123
456
789
,Des(ν)={4, 5, 7}
• Given a partition µ and a subset D ⊆ {(i , j) ∈ µ : i > 1},
ν(µ,D) = (ν(1), . . . , ν(k))
where k = µ1, ν(j) has size µ′j with cell contents {1, 2, . . . , µ′j} and
descent set Des(ν(j)) = {i : (i , j) ∈ D}.
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2 LLT polynomials indexed by vertical strips
1
2
3
4
5
6
7
8
9
10
11
⇐⇒
12
34
56
78
910
11
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2 LLT polynomials indexed by vertical strips
To read the inversion statistic:
7
8
6
5
2
2
9
9
1
1
6
⇐⇒
78
65
22
99
11
6
qq q
q
q qq
q9x21 x22 x5x
26 x7x8x
29
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Vertical-strip LLT’s are related to the Shuffle Conjecture
Shuffle Conjecture [HHLRU, ’2005]
⇒ Shuffle Theorem [Carlsson-Mellit, ’2017]
∇en = ∑σ∈WPn
qdinv(σ)tarea(σ)X σ
= ∑P
LLTP (X ; q),
where P’s are Schroder paths of size n.
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Vertical-strip LLT’s are generated by linear relations
Theorem [Alexandersson-Sulzgruber, ’2020]
There exists a unique function F : S → ΛQ(q), P 7→ FP , that satisfies
the following conditions:
1 Fndke = ek+1 for all k ∈N
2 FPQ = FPFQ for all P,Q ∈ S
3 (Unicellular relation)
UneV
−
UenV
= (q − 1)
UdV
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4 (Bounce relations)
Un.nVd .eW
= q ×
Un.nVe.dW
Ud .nVd .eW
=
Un.dVe.dW
Un.dVd .eW
= (q − 1)
Un.dVe.dW
+ q ×
Ud .nVe.dW20
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Note on vertical-strip LLT polynomials
• Vertical-strip LLT polynomials can be generated by linear relations
[Alexandersson-Sulzgruber, ’2020].
• As a result, e-positivity after the substitution q → q + 1 has been
obtained.
• LLT polynomials indexed by certain vertical strips are
Hall-Littlewood polynomials, after a little bit of modification.
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3 Unicellular LLT polynomials
1
2
3
4
5
6
7
⇐⇒
1
2
3
4
5
6
7
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To read the inversion statistic:
4
1
5
2
6
3
7
⇐⇒
4
1
5
2
6
3
7
q
q
q
q3x1x2 · · · x7
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Unicellular LLT’s are related to chromatic quasisymmetric func-
tions
f (x1, x2, . . . ) is quasisymmetric if for any a1, . . . , ak ∈ Z+,
[xa1i1 · · · xakik]f = [xa1j1 · · · x
akjk]f
whenever i1 < · · · < ik and j1 < · · · < jk .
Definition. For a simple graph G = (V ,E ), with V ⊂ Z+, the
chromatic quasisymmetric function is
XG (x ; q) = ∑κ, proper
qasc(κ)xκ,
where asc(κ) = |{{i , j} ∈ E : i < j and κ(i) < κ(j)}|.
3 4
2
1
6
5
1
4
2
3
34Example.
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Natural unit interval order and e-positivity conjecture
For a list of positive integers m = (m1,m2, . . . ,mn−1) satisfying
m1 ≤ m2 ≤ · · · ≤ mn−1 ≤ n and mi ≥ i , for all i ,
the natural unit interval order P(m) is the poset on [n] with the order
relation given by
i <P(m) j if i < n and mi + 1 ≤ j ≤ n.
Conjecture. [Shareshian-Wachs ’2016]
If G is the incomparability graph of a natural unit interval order,
then
XG (x ; q) is e-positive.
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Dyck path correspondence
In case of the natural unit interval order, the incomparability graph has a
Dyck path correspondence.
Example.
1 2
3
4 5
⇐⇒5
4
3
2
1
45
34
24 23
12
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Dyck path correspondence
In case of the natural unit interval order, the incomparability graph has a
Dyck path correspondence.
Example.
1 2
3
4 52 1 2 3
2⇐⇒
3
2
3
1
2
q
·q q
·
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Relation via plethysm
Proposition. [Carlsson-Mellit ’2017]
LLTν[(q − 1)X ; q] = (q − 1)nXGν(x ; q),
where the square bracket implies the plethystic substitution.
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Linear relation
Let G = (V ,E ) be a graph containing a triangle with certain
conditions. Then
Lemma. (q-version of triple-deletion property)
XG (x ; q) + qXG−{e1,e2}(x ; q) = (1 + q)XG−{e1}(x ; q).
The LLT correspondence :
Lemma. [Lee, ’2018+] (Local linear relation)
LLT+ q· LLT = (1 + q)· LLT
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Note on the linear relation
Lee’s linear relation is equivalent to the first bounce relation
Un.nVd .eW
= q ×
Un.nVe.dW
⇐⇒ = q ×
which can be proved by using the relation
− = (q − 1) ×
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Complete graph; Kn
The LLT diagram corresponding to the complete graph Kn is
⇐⇒
6
12
34
5
1
2
3
4
5
6area
a1 = 5
a2 = 4
a3 = 3
a4 = 2
a5 = 1
a6 = 0
LLTKn (x ; q) = H(n)(X ; q, t) = H(1n)(X ; q)
= ∑λ`n
∑T∈SYT(λ)
qcocharge(T )
sλ(x)
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Schur positivity of LLT
polynomials
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Schur positivity results
• [Grojnowski-Haiman, ’2006 (unpublished work)]
Proved Schur positivity using Hecke algebras and representation
theory.
• [J. Blasiak, ’2016]; in the case of the tuple consists of k ≤ 3
ribbons, explicit Schur expansion is given (using Kazhdan-Lusztig
theory, Lam-algebra and Fomin-Greene machinery).
Note. Thus this result proves the positivity of the q, t-Kostka
polynomials Kλµ(q, t) when µ has three columns.
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Unicellular case
• [Huh-Nam-Y., ’2020] In the case of LLT diagrams corresponding to
the class of melting lollipop graphs, explicit combinatorial formula
is given.
1 2 3 4 5 6
7 8
9
1011
• [Lee, ’2018] LLT polynomials with bandwidth ≤ 2 is proved to be
2-Schur positive.
• [Miller, ’2019] LLT polynomials with bandwidth ≤ 3 is proved to be
3-Schur positive.
Note. k-Schur positivity is stronger than Schur positivity.
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Conjecture
Conjecture. [Leclerc-Thibon, ’2000]
There is a statistic ca(T ) on standard Young tableaux depending
on the area sequence a such that
LLTa(X ; q) = ∑λ`n
∑T∈SYT(λ)
qca(T )
sλ(x)
which satisfies
∑κ∈Snproper
qasca(κ) = ∑λ`n
f λ ∑T∈SYT(λ)
qca(T )
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Remark
• [Alexandersson-Sulzgruber, ’2018] In the unicellular case,
power-sum expansion is known.
• [Alexandersson-Sulzgruber, ’2020] Unicellular and vertical-strip case,
after substituting q 7→ q + 1, e-expansion is proved.
• [Novelli-Thibon, ’2019] introduced a non-commutative lift of the
unicellular LLT polynomials and prove that they are positive in a
non-commutative analogue of the Gessel quasisymmetric basis, and
the relation to chromatic quasisymmetric function via
(non-commutative) plethysm.
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Wrap-up
• Definition: the LLT polynomial indexed by ν = (ν(1), . . . , ν(k)) is
LLTν(x ; q) = ∑T∈SSYT(ν)
qinv(T )xT .
• We are interested in finding combinatorial description for the Schur
coefficients.
• When the LLT polynomials are indexed by tuples of vertical strips
and unicellular diagrams, we can use corresponding paths model
and linear relations.
• In the unicellular LLT case, the Schur positivity is closely related to
the e-positivity conjecture of Shareshian and Wachs.
T H A N KY O U
36
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Wrap-up
• Definition: the LLT polynomial indexed by ν = (ν(1), . . . , ν(k)) is
LLTν(x ; q) = ∑T∈SSYT(ν)
qinv(T )xT .
• We are interested in finding combinatorial description for the Schur
coefficients.
• When the LLT polynomials are indexed by tuples of vertical strips
and unicellular diagrams, we can use corresponding paths model
and linear relations.
• In the unicellular LLT case, the Schur positivity is closely related to
the e-positivity conjecture of Shareshian and Wachs.
T H A N KY O U
36