The Kronecker product and the partition algebra Christopher Bowman - - PowerPoint PPT Presentation

▶
the kronecker product and the partition algebra
SMART_READER_LITE
LIVE PREVIEW

The Kronecker product and the partition algebra Christopher Bowman - - PowerPoint PPT Presentation

Oct 15, 2022 316 likes •935 views

The Kronecker product and the partition algebra Christopher Bowman Maud De Visscher Rosa Orellana FPSAC13 Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 1 / 14 The Kronecker problem Bowman,De

slide-1
SLIDE 1

The Kronecker product and the partition algebra

Christopher Bowman Maud De Visscher Rosa Orellana FPSAC’13

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 1 / 14

slide-2
SLIDE 2

The Kronecker problem

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

slide-3
SLIDE 3

The Kronecker problem Complex representations of GLn: simple (Weyl) modules V(λ).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

slide-4
SLIDE 4

The Kronecker problem Complex representations of GLn: simple (Weyl) modules V(λ). V(λ) ⊗ V(µ) =

  • ν

cν

λ,µV(ν),

where cν

λ,µ are the Littlewood-Richardson coefficients.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

slide-5
SLIDE 5

The Kronecker problem Complex representations of GLn: simple (Weyl) modules V(λ). V(λ) ⊗ V(µ) =

  • ν

cν

λ,µV(ν),

where cν

λ,µ are the Littlewood-Richardson coefficients.

Complex representations of Sn: simple (Specht) modules S(λ).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

slide-6
SLIDE 6

The Kronecker problem Complex representations of GLn: simple (Weyl) modules V(λ). V(λ) ⊗ V(µ) =

  • ν

cν

λ,µV(ν),

where cν

λ,µ are the Littlewood-Richardson coefficients.

Complex representations of Sn: simple (Specht) modules S(λ). S(λ) ⊗ S(µ) =

  • ν

gν

λ,µS(ν),

where gν

λ,µ are the Kronecker coefficients.

Combinatorial description of gν

λ,µ?

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 2 / 14

slide-7
SLIDE 7

Schur-Weyl dualities

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

slide-8
SLIDE 8

Schur-Weyl dualities

Let Vn be an n-dimensional C-vector space and r ≥ 1. Then we have the following Schur-Weyl dualities

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

slide-9
SLIDE 9

Schur-Weyl dualities

Let Vn be an n-dimensional C-vector space and r ≥ 1. Then we have the following Schur-Weyl dualities GLn → V ⊗r ← CSr

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

slide-10
SLIDE 10

Schur-Weyl dualities

Let Vn be an n-dimensional C-vector space and r ≥ 1. Then we have the following Schur-Weyl dualities GLn → V ⊗r ← CSr ∪ ∩ Sn Pr(n) Partition algebra

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

slide-11
SLIDE 11

Schur-Weyl dualities

Let Vn be an n-dimensional C-vector space and r ≥ 1. Then we have the following Schur-Weyl dualities GLn → V ⊗r ← CSr ∪ ∩ Sn Pr(n) Partition algebra Idea: Use the partition algebra to study the Kronecker problem.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 3 / 14

slide-12
SLIDE 12

Structure of the talk

1

The partition algebra Pr(n): Definition and first properties.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 4 / 14

slide-13
SLIDE 13

Structure of the talk

1

The partition algebra Pr(n): Definition and first properties.

2

Combinatorial representation theory of Pr(n).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 4 / 14

slide-14
SLIDE 14

Structure of the talk

1

The partition algebra Pr(n): Definition and first properties.

2

Combinatorial representation theory of Pr(n).

3

Application to the Kronecker problem.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 4 / 14

slide-15
SLIDE 15
  • 1. The partition algebra Pr(n)

Definition and first properties

Let r ∈ Z>0 and n ∈ C.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

slide-16
SLIDE 16
  • 1. The partition algebra Pr(n)

Definition and first properties

Let r ∈ Z>0 and n ∈ C. Pr(n) : C-algebra with basis given by all set partitions of {1, 2, . . . , r, 1′, 2′, . . . , r ′}.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

slide-17
SLIDE 17
  • 1. The partition algebra Pr(n)

Definition and first properties

Let r ∈ Z>0 and n ∈ C. Pr(n) : C-algebra with basis given by all set partitions of {1, 2, . . . , r, 1′, 2′, . . . , r ′}. {{1, 2, 4, 3′}, {3}, {5, 1′, 2′}, {4′}, {5′}}

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

slide-18
SLIDE 18
  • 1. The partition algebra Pr(n)

Definition and first properties

Let r ∈ Z>0 and n ∈ C. Pr(n) : C-algebra with basis given by all set partitions of {1, 2, . . . , r, 1′, 2′, . . . , r ′}. {{1, 2, 4, 3′}, {3}, {5, 1′, 2′}, {4′}, {5′}} ↔ 1′ 2′ 3′ 4′ 5′ 1 2 3 4 5

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 5 / 14

slide-19
SLIDE 19

and multiplication given by concatenation and scalar multiplication by nt where t is the number of connected components consisting of middle vertices only.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

slide-20
SLIDE 20

and multiplication given by concatenation and scalar multiplication by nt where t is the number of connected components consisting of middle vertices only. X =

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

slide-21
SLIDE 21

and multiplication given by concatenation and scalar multiplication by nt where t is the number of connected components consisting of middle vertices only. X = Y =

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

slide-22
SLIDE 22

and multiplication given by concatenation and scalar multiplication by nt where t is the number of connected components consisting of middle vertices only. X = Y = XY = n Remark: The group algebra CSr appears naturally as a subalgebra of Pr(n) (as the span of all diagrams having precisely r ‘propagating blocks’).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 6 / 14

slide-23
SLIDE 23

Assume throughout this talk that n = 0. Write Pr = Pr(n).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

slide-24
SLIDE 24

Assume throughout this talk that n = 0. Write Pr = Pr(n). e = 1

n

. . . e2 = e.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

slide-25
SLIDE 25

Assume throughout this talk that n = 0. Write Pr = Pr(n). e = 1

n

. . . e2 = e. ePre ∼ = Pr−1, Pr/PrePr ∼ = CSr.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

slide-26
SLIDE 26

Assume throughout this talk that n = 0. Write Pr = Pr(n). e = 1

n

. . . e2 = e. ePre ∼ = Pr−1, Pr/PrePr ∼ = CSr. Let L be a simple Pr-module. Then either eL = 0 and so L is a simple CSr-module, or eL = 0 and so eL is a simple Pr−1-module.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

slide-27
SLIDE 27

Assume throughout this talk that n = 0. Write Pr = Pr(n). e = 1

n

. . . e2 = e. ePre ∼ = Pr−1, Pr/PrePr ∼ = CSr. Let L be a simple Pr-module. Then either eL = 0 and so L is a simple CSr-module, or eL = 0 and so eL is a simple Pr−1-module. Thus we have that the simple Pr-modules are indexed by partitions of degree ≤ r.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 7 / 14

slide-28
SLIDE 28

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-29
SLIDE 29

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ≤r = {λ = (λ1, λ2, λ3, . . .), λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ 0,

i λi ≤ r}.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-30
SLIDE 30

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ≤r = {λ = (λ1, λ2, λ3, . . .), λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ 0,

i λi ≤ r}.

For each λ ∈ Λ≤r we have a cell module ∆r(λ), obtained by ‘inflating’ the corresponding Specht module.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-31
SLIDE 31

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ≤r = {λ = (λ1, λ2, λ3, . . .), λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ 0,

i λi ≤ r}.

For each λ ∈ Λ≤r we have a cell module ∆r(λ), obtained by ‘inflating’ the corresponding Specht module. λ ⊢ r, ∆r(λ) = S(λ) Specht module.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-32
SLIDE 32

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ≤r = {λ = (λ1, λ2, λ3, . . .), λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ 0,

i λi ≤ r}.

For each λ ∈ Λ≤r we have a cell module ∆r(λ), obtained by ‘inflating’ the corresponding Specht module. λ ⊢ r, ∆r(λ) = S(λ) Specht module. λ ⊢ r − 1, ∆r(λ) = Pre ⊗Pr−1 S(λ).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-33
SLIDE 33

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ≤r = {λ = (λ1, λ2, λ3, . . .), λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ 0,

i λi ≤ r}.

For each λ ∈ Λ≤r we have a cell module ∆r(λ), obtained by ‘inflating’ the corresponding Specht module. λ ⊢ r, ∆r(λ) = S(λ) Specht module. λ ⊢ r − 1, ∆r(λ) = Pre ⊗Pr−1 S(λ). . . .

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-34
SLIDE 34

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ≤r = {λ = (λ1, λ2, λ3, . . .), λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ 0,

i λi ≤ r}.

For each λ ∈ Λ≤r we have a cell module ∆r(λ), obtained by ‘inflating’ the corresponding Specht module. λ ⊢ r, ∆r(λ) = S(λ) Specht module. λ ⊢ r − 1, ∆r(λ) = Pre ⊗Pr−1 S(λ). . . .

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-35
SLIDE 35

Pr is not a semisimple algebra in general but it is a cellular algebra (as defined by Graham-Lehrer). Λ≤r = {λ = (λ1, λ2, λ3, . . .), λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ 0,

i λi ≤ r}.

For each λ ∈ Λ≤r we have a cell module ∆r(λ), obtained by ‘inflating’ the corresponding Specht module. λ ⊢ r, ∆r(λ) = S(λ) Specht module. λ ⊢ r − 1, ∆r(λ) = Pre ⊗Pr−1 S(λ). . . . A complete set of non-isomorphic simple Pr-modules is given by {Lr(λ) := hd ∆r(λ), λ ∈ Λ≤r}.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 8 / 14

slide-36
SLIDE 36
  • 2. Combinatorial representation theory of Pr(n) (P

. Martin)

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 9 / 14

slide-37
SLIDE 37
  • 2. Combinatorial representation theory of Pr(n) (P

. Martin) Theorem: Pr(n) is semisimple ⇔ n / ∈ {0, 1, 2, . . . , 2r − 2}.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 9 / 14

slide-38
SLIDE 38
  • 2. Combinatorial representation theory of Pr(n) (P

. Martin) Theorem: Pr(n) is semisimple ⇔ n / ∈ {0, 1, 2, . . . , 2r − 2}. We now take n ∈ Z>0.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 9 / 14

slide-39
SLIDE 39
  • 2. Combinatorial representation theory of Pr(n) (P

. Martin) Theorem: Pr(n) is semisimple ⇔ n / ∈ {0, 1, 2, . . . , 2r − 2}. We now take n ∈ Z>0. Definition: Let λ, µ be partitions. We say that (µ, λ) form an n-pair and we write µ ֒ →n λ if µ ⊂ λ and λ/µ consists of a single row of boxes of which the last (rightmost) one has content n − |µ|.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 9 / 14

slide-40
SLIDE 40
  • 2. Combinatorial representation theory of Pr(n) (P

. Martin) Theorem: Pr(n) is semisimple ⇔ n / ∈ {0, 1, 2, . . . , 2r − 2}. We now take n ∈ Z>0. Definition: Let λ, µ be partitions. We say that (µ, λ) form an n-pair and we write µ ֒ →n λ if µ ⊂ λ and λ/µ consists of a single row of boxes of which the last (rightmost) one has content n − |µ|. Example: ((2, 1), (4, 1)) form a 6-pair (with 6 − |µ| = 3). −1 0 1 ⊂ −1 0 1 2 3

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 9 / 14

slide-41
SLIDE 41

Proposition: The set Λ≤r splits into maximal chains of n-pairs: λ(0) ֒ →n λ(1) ֒ →n λ(2) ֒ →n . . . ֒ →n λ(t) (where t depends on the chain).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 10 / 14

slide-42
SLIDE 42

Proposition: The set Λ≤r splits into maximal chains of n-pairs: λ(0) ֒ →n λ(1) ֒ →n λ(2) ֒ →n . . . ֒ →n λ(t) (where t depends on the chain). Each cell module ∆r(λ(i)) (0 ≤ i ≤ t − 1) has Loewy structure Lr(λ(i)) Lr(λ(i+1)) .

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 10 / 14

slide-43
SLIDE 43

Proposition: The set Λ≤r splits into maximal chains of n-pairs: λ(0) ֒ →n λ(1) ֒ →n λ(2) ֒ →n . . . ֒ →n λ(t) (where t depends on the chain). Each cell module ∆r(λ(i)) (0 ≤ i ≤ t − 1) has Loewy structure Lr(λ(i)) Lr(λ(i+1)) . In the Grothendieck group we have [Lr(λ(i))] =

t

  • j=i

(−1)j−i[∆r(λ(j)].

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 10 / 14

slide-44
SLIDE 44
  • 3. Application to the Kronecker problem

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 11 / 14

slide-45
SLIDE 45
  • 3. Application to the Kronecker problem

Back to Schur-Weyl duality: As a (Sn, Pr(n))-bimodule we have V ⊗r

n

=

  • S(λ) ⊗ Lr(λ>1)

where the sum is over all λ = (λ1, λ2, λ3, . . .) partitions of n with λ>1 = (λ2, λ3, . . .) ∈ Λ≤r.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 11 / 14

slide-46
SLIDE 46
  • 3. Application to the Kronecker problem

Back to Schur-Weyl duality: As a (Sn, Pr(n))-bimodule we have V ⊗r

n

=

  • S(λ) ⊗ Lr(λ>1)

where the sum is over all λ = (λ1, λ2, λ3, . . .) partitions of n with λ>1 = (λ2, λ3, . . .) ∈ Λ≤r. Theorem: Let λ, µ, ν ⊢ n with λ>1 ⊢ r and µ>1 ⊢ s then we have gν

λ,µ =

[Lr+s(ν>1)↓Pr⊗Ps: Lr(λ>1) ⊗ Ls(µ>1)] if ν>1 ∈ Λ≤r+s

  • therwise

= t

i=0(−1)i[∆r+s(η(i))↓Pr⊗Ps: Lr(λ>1) ⊗ Ls(µ>1)]

if ν>1 ∈ Λ≤r+s

  • therwise

where ν>1 = η(0) ֒ →n η(1) ֒ →n η(2) ֒ →n . . . ֒ →n η(t) is the chain containing ν>1.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 11 / 14

slide-47
SLIDE 47

Consequences:

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 12 / 14

slide-48
SLIDE 48

Consequences: Murnaghan’s stability property: As we increase the length of the first row of the partitions we have gν

λ,µ → gν>1 λ>1,µ>1

reduced Kronecker coefficients

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 12 / 14

slide-49
SLIDE 49

Consequences: Murnaghan’s stability property: As we increase the length of the first row of the partitions we have gν

λ,µ → gν>1 λ>1,µ>1

reduced Kronecker coefficients Example S(12) ⊗ S(12) = S(2)

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 12 / 14

slide-50
SLIDE 50

Consequences: Murnaghan’s stability property: As we increase the length of the first row of the partitions we have gν

λ,µ → gν>1 λ>1,µ>1

reduced Kronecker coefficients Example S(12) ⊗ S(12) = S(2) S(2, 1) ⊗ S(2, 1) = S(3) ⊕ S(2, 1) ⊕ S(13)

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 12 / 14

slide-51
SLIDE 51

Consequences: Murnaghan’s stability property: As we increase the length of the first row of the partitions we have gν

λ,µ → gν>1 λ>1,µ>1

reduced Kronecker coefficients Example S(12) ⊗ S(12) = S(2) S(2, 1) ⊗ S(2, 1) = S(3) ⊕ S(2, 1) ⊕ S(13) S(3, 1) ⊗ S(3, 1) = S(4) ⊕ S(3, 1) ⊕ S(2, 12) ⊕ S(22)

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 12 / 14

slide-52
SLIDE 52

Consequences: Murnaghan’s stability property: As we increase the length of the first row of the partitions we have gν

λ,µ → gν>1 λ>1,µ>1

reduced Kronecker coefficients Example S(12) ⊗ S(12) = S(2) S(2, 1) ⊗ S(2, 1) = S(3) ⊕ S(2, 1) ⊕ S(13) S(3, 1) ⊗ S(3, 1) = S(4) ⊕ S(3, 1) ⊕ S(2, 12) ⊕ S(22) Then for all n ≥ 4 we have S(n−1, 1)⊗S(n−1, 1) = S(n)⊕S(n−1, 1)⊕S(n−2, 12)⊕S(n−2, 2).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 12 / 14

slide-53
SLIDE 53

Murnaghan’s stability property follows directly from the fact that Pr(n) is semisimple for large n.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 13 / 14

slide-54
SLIDE 54

Murnaghan’s stability property follows directly from the fact that Pr(n) is semisimple for large n. Representation theoretic interpretation of reduced Kronecker coefficients as composition factors of restriction of cell modules for partitions algebra to Young subalgebras.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 13 / 14

slide-55
SLIDE 55

Murnaghan’s stability property follows directly from the fact that Pr(n) is semisimple for large n. Representation theoretic interpretation of reduced Kronecker coefficients as composition factors of restriction of cell modules for partitions algebra to Young subalgebras. Recover bounds for this stability (Brion).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 13 / 14

slide-56
SLIDE 56

Murnaghan’s stability property follows directly from the fact that Pr(n) is semisimple for large n. Representation theoretic interpretation of reduced Kronecker coefficients as composition factors of restriction of cell modules for partitions algebra to Young subalgebras. Recover bounds for this stability (Brion). Closed positive formula for gν

λ,µ (for large enough n) when one of

the labelling partition is either a 2-part or a hook partition (as a sum of products of LR coefficients). This improves on work by Ballantine-Orellana (2-part case) and Blasiak (hook case).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 13 / 14

slide-57
SLIDE 57

Murnaghan’s stability property follows directly from the fact that Pr(n) is semisimple for large n. Representation theoretic interpretation of reduced Kronecker coefficients as composition factors of restriction of cell modules for partitions algebra to Young subalgebras. Recover bounds for this stability (Brion). Closed positive formula for gν

λ,µ (for large enough n) when one of

the labelling partition is either a 2-part or a hook partition (as a sum of products of LR coefficients). This improves on work by Ballantine-Orellana (2-part case) and Blasiak (hook case).

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 13 / 14

slide-58
SLIDE 58

Murnaghan’s stability property follows directly from the fact that Pr(n) is semisimple for large n. Representation theoretic interpretation of reduced Kronecker coefficients as composition factors of restriction of cell modules for partitions algebra to Young subalgebras. Recover bounds for this stability (Brion). Closed positive formula for gν

λ,µ (for large enough n) when one of

the labelling partition is either a 2-part or a hook partition (as a sum of products of LR coefficients). This improves on work by Ballantine-Orellana (2-part case) and Blasiak (hook case). Note: All proofs are very elementary.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 13 / 14

slide-59
SLIDE 59

Murnaghan’s stability property follows directly from the fact that Pr(n) is semisimple for large n. Representation theoretic interpretation of reduced Kronecker coefficients as composition factors of restriction of cell modules for partitions algebra to Young subalgebras. Recover bounds for this stability (Brion). Closed positive formula for gν

λ,µ (for large enough n) when one of

the labelling partition is either a 2-part or a hook partition (as a sum of products of LR coefficients). This improves on work by Ballantine-Orellana (2-part case) and Blasiak (hook case). Note: All proofs are very elementary. THANK YOU

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 13 / 14

slide-60
SLIDE 60

Let λ[n], µ[n], ν[n] be partitions of n with |λ| = r, |µ| = s and |ν| = r + s − l. (i) Suppose ν[n] = (n − k, k) is a two-part partition. Then we have g(n−k,k)

λ[n],µ[n] = g(k) λ,µ =

  • l1,l2

l=l1+2l2

  • σ⊢l1

γ⊢l2

cλ

(r−l1−l2),σ,γcµ γ,σ,(s−l1−l2)

for all n ≥ min{|λ| + µ1 + k, |µ| + λ1 + k}. (ii) Suppose ν[n] = (n − k, 1k) is a hook partition. Then we have g(n−k,1k)

λ[n],µ[n]

= g(1k)

λ,µ =

  • l1,l2

l=l1+2l2

  • σ⊢l1

γ⊢l2

cλ

(1r−l1−l2),σ,γcµ γ,σ′,(1s−l1−l2)

for all n ≥ min{|λ| + |µ| + 1, |µ| + λ1 + k, |λ| + µ1 + k} and where σ′ denotes the transpose of σ.

Bowman,De Visscher,Orellana Kronecker product and Partition algebra June 2013 14 / 14