Idempotent generation in partition monoids James East University of - PowerPoint PPT Presentation

Submonoids of P n � � α ∈ P n : | A ∩ n ′ | = 1 ( ∀ A ∈ α ) T n = — full transformation semigroup ∈ T 5 � � = op T n n ∼ T ∗ — T ∗ n = α ∈ P n : | A ∩ n | = 1 ( ∀ A ∈ α ) � � α ∈ P n : | A ∩ n ′ | ≤ 1 and | A ∩ n | ≤ 1 ( ∀ A ∈ α ) I n = — symmetric inverse monoid (aka rook monoid) ∈ I 5 James East Idempotent generation in partition monoids

Submonoids of P n � � α ∈ P n : | A ∩ n ′ | = 1 ( ∀ A ∈ α ) T n = — full transformation semigroup ∈ T 5 � � = op T n n ∼ T ∗ — T ∗ n = α ∈ P n : | A ∩ n | = 1 ( ∀ A ∈ α ) � � α ∈ P n : | A ∩ n ′ | ≤ 1 and | A ∩ n | ≤ 1 ( ∀ A ∈ α ) I n = — symmetric inverse monoid (aka rook monoid) ∈ I 5 In many ways, P n is just like a transformation semigroup. James East Idempotent generation in partition monoids

Generators Proposition There is a factorization P n = T n I n T ∗ n . Consequently, P n = � s 1 , . . . , s n − 1 , e 1 , . . . , e n , t 1 , . . . , t n − 1 � . 1 i n s i = 1 i n 1 i n e i = t i = James East Idempotent generation in partition monoids

Presentations Theorem (Halverson and Ram, 2005; E, 2011) The partition monoid P n has presentation P n ∼ = � s 1 , . . . , s n − 1 , e 1 , . . . , e n , t 1 , . . . , t n − 1 : (R1—R16) � , where (R1) s 2 (R9) t 2 i = 1 i = t i (R2) s i s j = s j s i (R10) t i t j = t j t i (R3) s i s j s i = s j s i s j (R11) s i t j = t j s i (R4) e 2 i = e i (R12) s i s j t i = t j s i s j (R5) e i e j = e j e i (R13) t i s i = s i t i = t i (R6) s i e j = e j s i (R14) t i e j = e j t i (R7) s i e i = e i +1 s i (R15) t i e j t i = t i (R8) e i e i +1 s i = e i e i +1 (R16) e j t i e j = e j . James East Idempotent generation in partition monoids

Presentations Theorem (Halverson and Ram, 2005; E, 2011) The partition algebra P δ n has presentation n ∼ P δ = � s 1 , . . . , s n − 1 , e 1 , . . . , e n , t 1 , . . . , t n − 1 : (R1—R16) � , where (R1) s 2 (R9) t 2 i = 1 i = t i (R2) s i s j = s j s i (R10) t i t j = t j t i (R3) s i s j s i = s j s i s j (R11) s i t j = t j s i (R4) e 2 i = δ e i (R12) s i s j t i = t j s i s j (R5) e i e j = e j e i (R13) t i s i = s i t i = t i (R6) s i e j = e j s i (R14) t i e j = e j t i (R7) s i e i = e i +1 s i (R15) t i e j t i = t i (R8) e i e i +1 s i = e i e i +1 (R16) e j t i e j = e j . James East Idempotent generation in partition monoids

Presentations Theorem (E, 2011) The singular partition monoid P n \ S n has presentation P n \ S n ∼ = � e 1 , . . . , e n , t ij (1 ≤ i < j ≤ n ) : (R1—R10) � , where (R1) e 2 i = e i (R6) e k t ij e k = e k (R2) e i e j = e j e i (R7) t ij e k = e k t ij (R3) t 2 ij = t ij (R8) t ij t jk = t jk t ki = t ki t ij (R9) e k t ki e i t ij e j t jk e k = e k t kj e j t ji e i t ik e k (R4) t ij t kl = t kl t ij (R5) t ij e k t ij = t ij (R10) e k t ki e i t ij e j t jl e l t lk e k = e k t kl e l t li e i t ij e j t jk e k . 1 i n 1 i j n e i = t ij = James East Idempotent generation in partition monoids

Presentations Theorem (Kudryavtseva and Mazorchuk, 2006; see also Birman-Wenzl and Barcelo-Ram) The Brauer monoid B n has presentation B n ∼ = � s 1 , . . . , s n − 1 , u 1 , . . . , u n − 1 : (R1—R10) � , where (R1) s 2 i = 1 (R5) u i u j = u j u i (R9) s i u j u i = s j u i (R2) s i s j = s j s i (R6) s i u j = u j s i (R10) u i u j s i = u i s j . (R3) s i s j s i = s j s i s j (R7) s i u i = u i s i = u i (R4) u 2 i = u i (R8) u i u j u i = u i 1 1 i n i n s i = u i = James East Idempotent generation in partition monoids

Presentations Theorem (Maltcev and Mazorchuk, 2007) The singular Brauer monoid B n \ S n has presentation B n \ S n ∼ = � u ij (1 ≤ i < j ≤ n ) : (R1—R6) � , where (R1) u 2 ij = u ij (R4) u ij u ik u jk = u ij u jk (R2) u ij u kl = u kl u ij (R5) u ij u jk u kl = u ij u il u kl (R3) u ij u jk u ij = u ij (R6) u ij u kl u ik = u ij u jl u ik . 1 i j n u ij = James East Idempotent generation in partition monoids

Presentations Theorem (Borisavljevi´ c, Doˇ sen, Petri´ c, 2002; see also Jones, Kauffman, etc) The (singular) Temperley-Lieb monoid TL n has presentation TL n ∼ = � u 1 , . . . , u n − 1 : (R1—R3) � , where (R1) u 2 i = u i (R2) u i u j = u j u i (R3) u i u j u i = u i . 1 i n u i = James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . What is the smallest number of (idempotent) partitions required to generate P n \ S n ? James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . What is the smallest number of (idempotent) partitions required to generate P n \ S n ? i.e., What is the rank and idempotent rank , rank( P n \ S n ) and idrank( P n \ S n )? James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . What is the smallest number of (idempotent) partitions required to generate P n \ S n ? i.e., What is the rank and idempotent rank , rank( P n \ S n ) and idrank( P n \ S n )? How many (idempotent) generating sets of minimal size are there? James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . What is the smallest number of (idempotent) partitions required to generate P n \ S n ? i.e., What is the rank and idempotent rank , rank( P n \ S n ) and idrank( P n \ S n )? How many (idempotent) generating sets of minimal size are there? What about other ideals of P n ? James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . What is the smallest number of (idempotent) partitions required to generate P n \ S n ? i.e., What is the rank and idempotent rank , rank( P n \ S n ) and idrank( P n \ S n )? How many (idempotent) generating sets of minimal size are there? What about other ideals of P n ? How many idempotents does P n contain? James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . What is the smallest number of (idempotent) partitions required to generate P n \ S n ? i.e., What is the rank and idempotent rank , rank( P n \ S n ) and idrank( P n \ S n )? How many (idempotent) generating sets of minimal size are there? What about other ideals of P n ? How many idempotents does P n contain? What about infinite partition monoids P X ? James East Idempotent generation in partition monoids

Idempotent generation — questions So the singular parts of P n , B n , TL n are idempotent generated . . . What is the smallest number of (idempotent) partitions required to generate P n \ S n ? i.e., What is the rank and idempotent rank , rank( P n \ S n ) and idrank( P n \ S n )? How many (idempotent) generating sets of minimal size are there? What about other ideals of P n ? How many idempotents does P n contain? What about infinite partition monoids P X ? Same questions for B n and TL n . . . James East Idempotent generation in partition monoids

Number of idempotents — B n Theorem ( Dolinka, E, Evangelou, FitzGerald, Ham, Hyde, Loughlin, 2014) The number of idempotents in the Brauer monoid B n is equal to n ! � e n = µ 1 ! · · · µ n ! · 2 µ 2 · · · (2 k ) µ 2 k µ ⊢ n where k = ⌊ n / 2 ⌋ — i.e., n = 2 k or 2 k + 1. James East Idempotent generation in partition monoids

Number of idempotents — B n Theorem ( Dolinka, E, Evangelou, FitzGerald, Ham, Hyde, Loughlin, 2014) The number of idempotents in the Brauer monoid B n is equal to n ! � e n = µ 1 ! · · · µ n ! · 2 µ 2 · · · (2 k ) µ 2 k µ ⊢ n where k = ⌊ n / 2 ⌋ — i.e., n = 2 k or 2 k + 1. The numbers e n satisfy the recurrence e 0 = 1, e n = a 1 e n − 1 + a 2 e n − 2 + · · · + a n e 0 � n − 1 � � n − 1 � where a 2 i = (2 i − 1)! and a 2 i +1 = (2 i + 1)!. 2 i − 1 2 i James East Idempotent generation in partition monoids

Number of idempotents — B δ n Theorem (DEEFHHL, 2014) The number of idempotent basis elements in the Brauer algebra B δ n is equal to n ! � µ 1 ! µ 3 ! · · · µ 2 k +1 ! , µ where k = ⌊ n − 1 2 ⌋ , the sum is over all integer partitions µ ⊢ n with only odd parts, δ is not a root of unity. James East Idempotent generation in partition monoids

Number of idempotents — P n Theorem (DEEFHHL, 2014) The number of idempotents in the partition monoid P n is equal to c (1) µ 1 · · · c ( n ) µ n � n ! · µ 1 ! · · · µ n ! · (1!) µ 1 · · · ( n !) µ n , µ ⊢ n where k � c ( k ) = (1 + rs ) c ( k , r , s ), and r , s =1 c ( k , r , 1) = S ( k , r ) c ( k , 1 , s ) = S ( k , s ) c ( k , r , s ) = s · c ( k − 1 , r − 1 , s ) + r · c ( k − 1 , r , s − 1) + rs · c ( k − 1 , r , s ) k − 2 � r − 1 s − 1 � k − 2 � � � � � + a ( s − b ) + b ( r − a ) c ( m , a , b ) c ( k − m − 1 , r − a , s − b ) m m =1 a =1 b =1 if r , s ≥ 2. James East Idempotent generation in partition monoids

Number of idempotents — P δ n Theorem (DEEFHHL, 2014) The number of idempotent basis elements in the partition algebra P δ n is equal to c ′ (1) µ 1 · · · c ′ ( n ) µ n � n ! · µ 1 ! · · · µ n ! · (1!) µ 1 · · · ( n !) µ n , µ ⊢ n where k � c ′ ( k ) = rs · c ( k , r , s ), and r , s =1 δ is not a root of unity. James East Idempotent generation in partition monoids

Less algebra, more diagrams. . . James East Idempotent generation in partition monoids

Number of idempotents — TL 1 – TL 7 (GAP) The number of idempotents in TL n is currently unknown. James East Idempotent generation in partition monoids

Number of idempotents — TL 8 – TL 11 (GAP) The number of idempotents in TL n is currently unknown. James East Idempotent generation in partition monoids

Number of idempotents — inside TL 15 – TL 17 (GAP) The number of idempotents in TL n is currently unknown. Thanks to Attila Egri-Nagy for these . . . James East Idempotent generation in partition monoids

Rank and idempotent rank — P n \ S n James East Idempotent generation in partition monoids

Rank and idempotent rank — P n \ S n Theorem (E, 2011) P n \ S n is idempotent generated. P n \ S n = � e 1 , . . . , e n , t ij (1 ≤ i < j ≤ n ) � . 1 r n 1 i j n e r = t ij = James East Idempotent generation in partition monoids

Rank and idempotent rank — P n \ S n Theorem (E, 2011) P n \ S n is idempotent generated. P n \ S n = � e 1 , . . . , e n , t ij (1 ≤ i < j ≤ n ) � . 1 r n 1 i j n e r = t ij = � n � � n +1 � = n ( n +1) rank( P n \ S n ) = idrank( P n \ S n ) = n + = . 2 2 2 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Any minimal idempotent generating set for P n \ S n is a subset of { e r : 1 ≤ r ≤ n } ∪ { t ij , f ij , f ji , g ij , g ji : 1 ≤ i < j ≤ n } . 1 r n 1 i j n e r = t ij = 1 i j n 1 i j n f ij = f ji = 1 i j n 1 i j n g ij = g ji = James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Any minimal idempotent generating set for P n \ S n is a subset of { e r : 1 ≤ r ≤ n } ∪ { t ij , f ij , f ji , g ij , g ji : 1 ≤ i < j ≤ n } . 1 r n 1 i j n e r = t ij = 1 i j n 1 i j n f ij = f ji = 1 i j n 1 i j n g ij = g ji = To see which subsets generate P n \ S n , we create a graph. . . James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Let Γ n be the di-graph with vertex set V (Γ n ) = { A ⊆ n : | A | = 1 or | A | = 2 } and edge set E (Γ n ) = { ( A , B ) : A ⊆ B or B ⊆ A } . 1 15 12 5 2 25 14 13 45 35 24 23 4 3 34 Γ 5 (with loops omitted) James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n For only $59.95. . . James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Let Γ n be the di-graph with vertex set V (Γ n ) = { A ⊆ n : | A | = 1 or | A | = 2 } and edge set E (Γ n ) = { ( A , B ) : A ⊆ B or B ⊆ A } . 1 15 12 5 2 25 14 13 45 35 24 23 4 3 34 Γ 5 (with loops omitted) James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Let Γ n be the di-graph with vertex set V (Γ n ) = { A ⊆ n : | A | = 1 or | A | = 2 } and edge set E (Γ n ) = { ( A , B ) : A ⊆ B or B ⊆ A } . e 1 = 1 15 12 5 2 25 14 13 45 35 24 23 4 3 34 Γ 5 (with loops omitted) James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Let Γ n be the di-graph with vertex set V (Γ n ) = { A ⊆ n : | A | = 1 or | A | = 2 } and edge set E (Γ n ) = { ( A , B ) : A ⊆ B or B ⊆ A } . e 1 = 1 15 12 5 2 25 14 13 = t 45 45 35 24 23 4 3 34 Γ 5 (with loops omitted) James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Let Γ n be the di-graph with vertex set V (Γ n ) = { A ⊆ n : | A | = 1 or | A | = 2 } and edge set E (Γ n ) = { ( A , B ) : A ⊆ B or B ⊆ A } . e 1 = 1 15 12 5 2 25 14 13 = t 45 45 35 24 23 f 23 = 4 3 34 Γ 5 (with loops omitted) James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Let Γ n be the di-graph with vertex set V (Γ n ) = { A ⊆ n : | A | = 1 or | A | = 2 } and edge set E (Γ n ) = { ( A , B ) : A ⊆ B or B ⊆ A } . e 1 = 1 = g 51 15 12 5 2 25 14 13 = t 45 45 35 24 23 f 23 = 4 3 34 Γ 5 (with loops omitted) James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n A subgraph H of a di-graph G is a permutation subgraph if V ( H ) = V ( G ) and the edges of H induce a permutation of V ( G ). 1 15 12 5 2 25 14 13 45 35 24 23 4 34 3 A permutation subgraph of Γ n is determined by: a permutation of a subset A of n with no fixed points or 2-cycles ( A = { 2 , 3 , 5 } , 2 �→ 3 �→ 5 �→ 2), and a function n \ A → n with no 2-cycles (1 �→ 4, 4 �→ 4). James East Idempotent generation in partition monoids

Minimal idempotent generating sets — P n \ S n Theorem (E+Gray, 2014) The minimal idempotent generating sets of P n \ S n are in one-one correspondence with the permutation subgraphs of Γ n . The number of minimal idempotent generating sets of P n \ S n is equal to n � n � � a k b n , n − k , k k =0 where a 0 = 1, a 1 = a 2 = 0, a k +1 = ka k + k ( k − 1) a k − 2 , and ⌊ k 2 ⌋ � k � � ( − 1) i (2 i − 1)!! n k − 2 i . b n , k = 2 i i =0 n 0 1 2 3 4 5 6 7 · · · 1 1 3 20 201 2604 40915 754368 · · · James East Idempotent generation in partition monoids

Ideals — P n \ S n James East Idempotent generation in partition monoids

Ideals — P n \ S n The ideals of P n are I r = { α ∈ P n : α has ≤ r transverse blocks } for 0 ≤ r ≤ n . James East Idempotent generation in partition monoids

Ideals — P n \ S n The ideals of P n are I r = { α ∈ P n : α has ≤ r transverse blocks } for 0 ≤ r ≤ n . Theorem (E+Gray, 2014) If 0 ≤ r ≤ n − 1, then I r is idempotent generated, and n � j � � rank( I r ) = idrank( I r ) = S ( n , j ) . r j = r James East Idempotent generation in partition monoids

Ideals — P n \ S n The ideals of P n are I r = { α ∈ P n : α has ≤ r transverse blocks } for 0 ≤ r ≤ n . Theorem (E+Gray, 2014) If 0 ≤ r ≤ n − 1, then I r is idempotent generated, and n � j � � rank( I r ) = idrank( I r ) = S ( n , j ) . r j = r The idempotent generating sets of this size have not been classified/enumerated (for 1 ≤ r ≤ n − 2). James East Idempotent generation in partition monoids

Rank and idempotent rank — B n \ S n James East Idempotent generation in partition monoids

Rank and idempotent rank — B n \ S n Theorem (Maltcev and Mazorchuk, 2007) B n \ S n is idempotent generated. B n \ S n = � u ij (1 ≤ i < j ≤ n ) � . 1 i j n u ij = James East Idempotent generation in partition monoids

Rank and idempotent rank — B n \ S n Theorem (Maltcev and Mazorchuk, 2007) B n \ S n is idempotent generated. B n \ S n = � u ij (1 ≤ i < j ≤ n ) � . 1 i j n u ij = � n � = n ( n − 1) rank( B n \ S n ) = idrank( B n \ S n ) = . 2 2 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — B n \ S n Let Λ n be the di-graph with vertex set V (Λ n ) = { A ⊆ n : | A | = 2 } and edge set E (Λ n ) = { ( A , B ) : A ∩ B � = ∅} . 12 12 23 13 24 14 23 13 34 Λ 3 Λ 4 James East Idempotent generation in partition monoids

Minimal idempotent generating sets — B n \ S n Theorem (E+Gray, 2014) The minimal idempotent generating sets of B n \ S n are in one-one correspondence with the permutation subgraphs of Λ n . No formula is known for the number of minimal idempotent generating sets of B n \ S n (yet). Very hard! n 0 1 2 3 4 5 6 7 1 1 1 6 265 126,140 855,966,441 ???? 1 1 1 2 12 288 34,560 24,883,200 There are (way) more than ( n − 1)! · ( n − 2)! · · · 3! · 2! · 1!. Thanks to James Mitchell for n = 5 , 6 (GAP). James East Idempotent generation in partition monoids

Ideals — B n \ S n The ideals of B n are I r = { α ∈ B n : α has ≤ r transverse blocks } for 0 ≤ r = n − 2 k ≤ n . Theorem (E+Gray, 2014) If 0 ≤ r = n − 2 k ≤ n − 2, then I r is idempotent generated and � n � n ! rank( I r ) = idrank( I r ) = (2 k − 1)!! = 2 k k ! r ! . 2 k James East Idempotent generation in partition monoids

Rank and idempotent rank — TL n Theorem (Borisavljevi´ c, Doˇ sen, Petri´ c, 2002, etc) TL n is idempotent generated. TL n = � u 1 , . . . , u n − 1 � . 1 i n u i = rank( TL n ) = idrank( TL n ) = n − 1. James East Idempotent generation in partition monoids