Transition matrices for symmetric and quasisymmetric Hall-Littlewood - - PowerPoint PPT Presentation

▶
transition matrices for symmetric and quasisymmetric hall
SMART_READER_LITE
LIVE PREVIEW

Transition matrices for symmetric and quasisymmetric Hall-Littlewood - - PowerPoint PPT Presentation

Dec 07, 2023 123 likes •867 views

Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials Nick Loehr Virginia Tech and U.S. Naval Academy Luis Serrano LaCIM, UQ` AM Greg Warrington University of Vermont FPSAC SFCA Paris, France June

slide-1
SLIDE 1

Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials

Nick Loehr – Virginia Tech and U.S. Naval Academy Luis Serrano – LaCIM, UQ` AM Greg Warrington – University of Vermont FPSAC – SFCA Paris, France June 27, 2013

1 / 41

slide-2
SLIDE 2

Original motivation

Schur − → fundamental quasisymmetric s31 = F31 + F22 + F13

2 / 41

slide-3
SLIDE 3

Original motivation

Schur − → fundamental quasisymmetric s31 = F31 + F22 + F13 monomial symmetric − → monomial quasisymmetric m31 = M31 + M13

2 / 41

slide-4
SLIDE 4

Original motivation

Schur − → fundamental quasisymmetric s31 = F31 + F22 + F13 monomial symmetric − → monomial quasisymmetric m31 = M31 + M13 Hall-Littlewood polynomials Pλ(x; t) Pλ(x; 0) = sλ Pλ(x; 1) = mλ

2 / 41

slide-5
SLIDE 5

Original motivation

Schur − → fundamental quasisymmetric s31 = F31 + F22 + F13 monomial symmetric − → monomial quasisymmetric m31 = M31 + M13 Hall-Littlewood polynomials Pλ(x; t) Pλ(x; 0) = sλ Pλ(x; 1) = mλ Question: Is there some quasisymmetric expansion of Pλ which:

◮ at t = 0 gives us the fundamental expansion of Schur

functions, and

◮ at t = 1 gives us the monomial quasisymmetric expansion of

monomial symmetric functions?

2 / 41

slide-6
SLIDE 6

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions t=0 t=0 t=1 t=1

3 / 41

slide-7
SLIDE 7

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

4 / 41

slide-8
SLIDE 8

Symmetric functions

monomial: m421 =

  • i,j,k

x4

i x2 j x1 k

5 / 41

slide-9
SLIDE 9

Symmetric functions

monomial: m421 =

  • i,j,k

x4

i x2 j x1 k

Schur: s21 = x2

1x2

+ x1x2

2

+ x1x2x3 + x1x2x3 · · · 2 1 1 2 1 2 3 1 2 2 1 3

5 / 41

slide-10
SLIDE 10

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions SSYT

6 / 41

slide-11
SLIDE 11

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

(E˘ gecio˘ glu – Remmel)

SRHT

7 / 41

slide-12
SLIDE 12

Quasisymmetric functions

Monomial: M142 =

  • i<j<k

x1

i x4 j x2 k

8 / 41

slide-13
SLIDE 13

Quasisymmetric functions

Monomial: M142 =

  • i<j<k

x1

i x4 j x2 k

Fundamental (refinement order – Gessel): F23 = M23 + M221 + M212 + M2111 + M113 + M1121 + M1112 + M11111

8 / 41

slide-14
SLIDE 14

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

(Gessel)

refinement

9 / 41

slide-15
SLIDE 15

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions inversion M¨

  • bius

10 / 41

slide-16
SLIDE 16

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

(Gessel)

descents unsort

11 / 41

slide-17
SLIDE 17

Symmetric to quasisymmetric

Schur to fundamental, sum over descent compositions (Gessel) s31 = F31 + F22 + F13 4 1 2 3 3 1 2 4 2 1 3 4

12 / 41

slide-18
SLIDE 18

Symmetric to quasisymmetric

Schur to fundamental, sum over descent compositions (Gessel) s31 = F31 + F22 + F13 4 1 2 3 3 1 2 4 2 1 3 4 Monomial symmetric to monomial quasisymmetric (unsort) m321 = M321 + M312 + M213 + M231 + M132 + M123

12 / 41

slide-19
SLIDE 19

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

(Egge-Loehr- Warrington, Garsia)

sort

13 / 41

slide-20
SLIDE 20

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13?

14 / 41

slide-21
SLIDE 21

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

14 / 41

slide-22
SLIDE 22

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream) 14 / 41

slide-23
SLIDE 23

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream)

What is s13????

14 / 41

slide-24
SLIDE 24

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream)

What is s13???? s13 = −s22.

14 / 41

slide-25
SLIDE 25

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream)

What is s13???? s13 = −s22. What is sα if α is not a partition?

14 / 41

slide-26
SLIDE 26

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream)

What is s13???? s13 = −s22. What is sα if α is not a partition? s154 = s433

14 / 41

slide-27
SLIDE 27

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream)

What is s13???? s13 = −s22. What is sα if α is not a partition? s154 = s433 s154 = (−1)2s433

14 / 41

slide-28
SLIDE 28

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream)

What is s13???? s13 = −s22. What is sα if α is not a partition? s154 = s433 s154 = (−1)2s433

  • 14 / 41
slide-29
SLIDE 29

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

Schur expansion of F31 + F22 + F13? F31 + F22 + F13 = s31 + s22 + s13

(First year graduate student’s dream)

What is s13???? s13 = −s22. What is sα if α is not a partition? s154 = s433 s154 = (−1)2s433

  • 14 / 41
slide-30
SLIDE 30

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

s12 =

15 / 41

slide-31
SLIDE 31

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

s12 =

  • 15 / 41
slide-32
SLIDE 32

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

s13 = −s22

  • 16 / 41
slide-33
SLIDE 33

Quasisymmetric to symmetric (Egge-Loehr-Warrington, Garsia)

s13 = −s22

  • F31 + F22 + F13

= s31 + s22 + s13 = s31 + s22 − s22 = s31

16 / 41

slide-34
SLIDE 34

Moral of this talk

17 / 41

slide-35
SLIDE 35

18 / 41

slide-36
SLIDE 36

Moral

Function Schur Symmetric Functions Quasisymmetric Functions hard

19 / 41

slide-37
SLIDE 37

(Counterexample to the triangle inequality)

Function Schur Fundamental Symmetric Functions Quasisymmetric Functions hard easy

(easy)

E-L-W,G

20 / 41

slide-38
SLIDE 38

(Counterexample to the triangle inequality)

Function Schur Fundamental Symmetric Functions Quasisymmetric Functions hard easy

(easy)

E-L-W,G

Side effects may include hallucinations, such as an apparent loss of Schur positivity. 20 / 41

slide-39
SLIDE 39

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

(Carbonara)

Tournament matrices

(Macdonald)

algebraic

21 / 41

slide-40
SLIDE 40

Prism of bases

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

(Lascoux-Sch¨ utzenberger)

charge

22 / 41

slide-41
SLIDE 41

New Transition matrices (LSW)

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

23 / 41

slide-42
SLIDE 42

Hall-Littlewood to Fundamental

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

24 / 41

slide-43
SLIDE 43

Hall-Littlewood polynomials

The Pλ(x; t) satisfy: Pλ(x; 0) = sλ(x) Pλ(x; 1) = mλ(x)

25 / 41

slide-44
SLIDE 44

Hall-Littlewood polynomials

The Pλ(x; t) satisfy: Pλ(x; 0) = sλ(x) Pλ(x; 1) = mλ(x)

Theorem (Loehr-S.-Warrington)

Pλ/µ(x; t) =

  • S∗∈SYT∗(λ/µ)

sgn(S∗)ttstat(S∗)FAsc′(S∗)(x).

25 / 41

slide-45
SLIDE 45

Hall-Littlewood polynomials

The Pλ(x; t) satisfy: Pλ(x; 0) = sλ(x) Pλ(x; 1) = mλ(x)

Theorem (Loehr-S.-Warrington)

Pλ/µ(x; t) =

  • S∗∈SYT∗(λ/µ)

sgn(S∗)ttstat(S∗)FAsc′(S∗)(x). P21(t) = F21 − tF111 + F12 − t2F111. 3 1 2 3 1 2∗ 2 1 3 2 1 3∗

25 / 41

slide-46
SLIDE 46

Starred tableaux

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

26 / 41

slide-47
SLIDE 47

Starred tableaux

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

Ascents:

◮ When i + 1 is above i (or i∗).

26 / 41

slide-48
SLIDE 48

Starred tableaux

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

Ascents:

◮ When i + 1 is above i (or i∗). ◮ When (i + 1)∗ is in the next column to the right of i (or i∗).

26 / 41

slide-49
SLIDE 49

Starred tableaux

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

Ascents:

◮ When i + 1 is above i (or i∗). ◮ When (i + 1)∗ is in the next column to the right of i (or i∗).

Asc(S∗) = {3, 4, 6, 7, 9, 10, 14}

26 / 41

slide-50
SLIDE 50

Starred tableaux

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

Ascents:

◮ When i + 1 is above i (or i∗). ◮ When (i + 1)∗ is in the next column to the right of i (or i∗).

Asc(S∗) = {3, 4, 6, 7, 9, 10, 14} 3 − 0

3

, 4 − 1

1

, 6 − 4

2

, 7 − 6

1

, 9 − 7

2

, 10 − 9

1

, 14 − 10

4

, 15 − 14

1

26 / 41

slide-51
SLIDE 51

Starred tableaux

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

Ascents:

◮ When i + 1 is above i (or i∗). ◮ When (i + 1)∗ is in the next column to the right of i (or i∗).

Asc(S∗) = {3, 4, 6, 7, 9, 10, 14} 3 − 0

3

, 4 − 1

1

, 6 − 4

2

, 7 − 6

1

, 9 − 7

2

, 10 − 9

1

, 14 − 10

4

, 15 − 14

1

Asc′(S∗) = 31212141

26 / 41

slide-52
SLIDE 52

Starred tableaux t-statistic

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

tstat(S∗) =

27 / 41

slide-53
SLIDE 53

Starred tableaux t-statistic

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

tstat(S∗) = 1

28 / 41

slide-54
SLIDE 54

Starred tableaux t-statistic

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

tstat(S∗) = 1 + 2

29 / 41

slide-55
SLIDE 55

Starred tableaux t-statistic

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

tstat(S∗) = 1 + 2 + 1

30 / 41

slide-56
SLIDE 56

Starred tableaux t-statistic

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

tstat(S∗) = 1 + 2 + 1 + 2 = 6

31 / 41

slide-57
SLIDE 57

Starred tableaux expansion

λ = 65211 S∗ = 11 10 5 12 4 6 7∗ 13

∗ 14

1 2 3 8∗ 9 15

∗

sgn(S∗) = (−1)4 , Asc′(S∗) = 31212141 , tstat(S∗) = 6 S∗ contributes to P65211(x; t) a term of the form sgn(S∗)ttstat(S∗)FAsc′(S∗) = t6F31212141.

32 / 41

slide-58
SLIDE 58

Schur expansion of Hall-Littlewood polynomials

Fundamental expansion of Hall–Littlewood P21(t) = F21 − tF111 + F12 − t2F111. 3 1 2 3 1 2∗ 2 1 3 2 1 3∗

33 / 41

slide-59
SLIDE 59

Schur expansion of Hall-Littlewood polynomials

Fundamental expansion of Hall–Littlewood P21(t) = F21 − tF111 + F12 − t2F111. 3 1 2 3 1 2∗ 2 1 3 2 1 3∗ Schur expansion of Hall–Littlewood P21(t) = s21 − ts111 + s12 − t2s111.

33 / 41

slide-60
SLIDE 60

Schur expansion of Hall-Littlewood polynomials

Fundamental expansion of Hall–Littlewood P21(t) = F21 − tF111 + F12 − t2F111. 3 1 2 3 1 2∗ 2 1 3 2 1 3∗ Schur expansion of Hall–Littlewood P21(t) = s21 − ts111 + s12 − t2s111. = s21 − ts111 + − t2s111.

33 / 41

slide-61
SLIDE 61

Schur expansion of Hall-Littlewood polynomials

Fundamental expansion of Hall–Littlewood P21(t) = F21 − tF111 + F12 − t2F111. 3 1 2 3 1 2∗ 2 1 3 2 1 3∗ Schur expansion of Hall–Littlewood P21(t) = s21 − ts111 + s12 − t2s111. = s21 − ts111 + − t2s111. = s21 − (t + t2) s111

33 / 41

slide-62
SLIDE 62

Hivert to Fundamental and Monomial

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

(Hivert)

s

(Hivert)

Bre

34 / 41

slide-63
SLIDE 63

Hivert quasisymmetric functions

For γ = (γ1, . . . , γp) a composition, Gγ(x1, ..., xn; t) = 1 [p]t![n − p]t! ⊡ω (xγ1

1 · · · xγp p ).

⊡ω is a t-symmetrizing operator.

35 / 41

slide-64
SLIDE 64

Hivert quasisymmetric functions

For γ = (γ1, . . . , γp) a composition, Gγ(x1, ..., xn; t) = 1 [p]t![n − p]t! ⊡ω (xγ1

1 · · · xγp p ).

⊡ω is a t-symmetrizing operator.

Theorem (Hivert)

Gα(x; 0) = Fα(x) and Gα(x; 1) = Mα(x)

35 / 41

slide-65
SLIDE 65

Fundamental and Monomial to Hivert

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

36 / 41

slide-66
SLIDE 66

Hivert expansions

Theorem (Loehr-S.-Warrington)

M(F, G)γ,β =

  • tg(γ,β),

if β γ, 0, else.

Theorem (Loehr-S.-Warrington)

M(M, G)α,β = (−1)ℓ(β)−ℓ(α)

  • j: ξα,β(j)=j

(1 − tj).

◮ ξγ,β(j) is j if βj and βj+1 contribute to the same part of γ and

0 otherwise.

◮ g(γ, β) = ℓ(β)−1 j=1

ξγ,β(j).

37 / 41

slide-67
SLIDE 67

Hall-Littlewood to Hivert

Hall-Littlewood Schur monomial Hivert Monomial Fundamental Symmetric Functions Quasisymmetric Functions

38 / 41

slide-68
SLIDE 68

Hivert expansion of Hall-Littlewood polynomials

Theorem (Loehr-S.-Warrington)

Pλ =

  • S∈SYT(λ)

Des(S)⊆sub(β)

Gβ   

  • j∈sub(β):

cj+1∈Esp(S)

  • tmj − twt(cj+1)

j∈[n−1]: cj+1∈Sp(S)\Esp(S)

tm′

j

  • 1 − twt(cj+1)

  

39 / 41

slide-69
SLIDE 69

Hivert expansion of Hall-Littlewood polynomials

Theorem (Loehr-S.-Warrington)

Pλ =

  • S∈SYT(λ)

Des(S)⊆sub(β)

Gβ   

  • j∈sub(β):

cj+1∈Esp(S)

  • tmj − twt(cj+1)

j∈[n−1]: cj+1∈Sp(S)\Esp(S)

tm′

j

  • 1 − twt(cj+1)

   P31 = G31 + (1 − t)G22 + (t2 − t3)G211 + G13 + (t2 − t)G121

39 / 41

slide-70
SLIDE 70

Hivert expansion of Hall-Littlewood polynomials

Theorem (Loehr-S.-Warrington)

Pλ =

  • S∈SYT(λ)

Des(S)⊆sub(β)

Gβ   

  • j∈sub(β):

cj+1∈Esp(S)

  • tmj − twt(cj+1)

j∈[n−1]: cj+1∈Sp(S)\Esp(S)

tm′

j

  • 1 − twt(cj+1)

   P31 = G31 + (1 − t)G22 + (t2 − t3)G211 + G13 + (t2 − t)G121 Plugging t = 0, get s31 = F31 + F22 + F13

39 / 41

slide-71
SLIDE 71

Hivert expansion of Hall-Littlewood polynomials

Theorem (Loehr-S.-Warrington)

Pλ =

  • S∈SYT(λ)

Des(S)⊆sub(β)

Gβ   

  • j∈sub(β):

cj+1∈Esp(S)

  • tmj − twt(cj+1)

j∈[n−1]: cj+1∈Sp(S)\Esp(S)

tm′

j

  • 1 − twt(cj+1)

   P31 = G31 + (1 − t)G22 + (t2 − t3)G211 + G13 + (t2 − t)G121 Plugging t = 0, get s31 = F31 + F22 + F13 Plugging t = 1, get m31 = M31 + M13

39 / 41

slide-72
SLIDE 72

Merci beaucoup

40 / 41

slide-73
SLIDE 73

Full version:

  • N. Loehr, L. Serrano, G. Warrington, Transition matrices for

symmetric and quasisymmetric Hall-Littlewood polynomials To appear in Journal of Combinatorial Theory Series A http://arxiv.org/abs/1202.3411 Slides at: http://www.thales.math.uqam.ca/˜ serrano/slides.html Sage code at: http://www.cems.uvm.edu/˜ gswarrin/

41 / 41