Leaf poset and multi-colored hook length property . . . Masao - PowerPoint PPT Presentation

. (Shifted) diagrams . . Definition . . A partiton is a nonincreasing sequence λ = ( λ 1 , λ 2 , . . . ) of nonnegative integers with finitely many λ i unequal to zero. The length and weight of λ , denoted by ℓ ( λ ) and | λ | , are the number and sum of the non-zero λ i respectively. A strict partition is a partition in which its parts are strictly decreasing. If λ is a partition (resp. strict partition), then its diagram D ( λ ) (resp. shifted diagram S ( λ ) ) is defined by D ( λ ) = { ( i , j ) ∈ Z 2 : 1 ≤ j ≤ λ i } S ( λ ) = { ( i , j ) ∈ Z 2 : i ≤ j ≤ λ i + i − 1 } . . . . . Example (The diagram and shifted diagram for λ = ( 4 , 3 , 1 ) ) . D ( λ ) = S ( λ ) = . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. (Shifted) diagrams . . Definition . . We define the order on D ( λ ) (or S ( λ ) ) by ( i 1 , j 1 ) ≥ ( i 2 , j 2 ) ⇔ i 1 ≤ i 2 and j 1 ≤ j 2 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. (Shifted) diagrams . . Definition . . We define the order on D ( λ ) (or S ( λ ) ) by ( i 1 , j 1 ) ≥ ( i 2 , j 2 ) ⇔ i 1 ≤ i 2 and j 1 ≤ j 2 We rotate the Hasse diagram of the poset by 45 ◦ counterclockwise. Hence a vertex in the north-east is bigger than a vertex in south-west. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Examples . shape − → shifted shape − → . . . . . . Masao Ishikawa Leaf poset and hook length property

d -complete poset . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . d -complete poset . . Contents of this section . . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . d -complete poset . . Contents of this section . . . The d -complete posets arise from the dominant 1 minuscule heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . d -complete poset . . Contents of this section . . . The d -complete posets arise from the dominant 1 minuscule heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras. . . Proctor gave completely combinatorial description of 2 d -complete poset, which is a graded poset with d -complete coloring. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . d -complete poset . . Contents of this section . . . The d -complete posets arise from the dominant 1 minuscule heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras. . . Proctor gave completely combinatorial description of 2 d -complete poset, which is a graded poset with d -complete coloring. . . Proctor showed that any d -complete poset can be 3 obtained from the 15 irreducible classes by slant-sum . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . d -complete poset . . Contents of this section . . . The d -complete posets arise from the dominant 1 minuscule heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras. . . Proctor gave completely combinatorial description of 2 d -complete poset, which is a graded poset with d -complete coloring. . . Proctor showed that any d -complete poset can be 3 obtained from the 15 irreducible classes by slant-sum . . . The d -complete coloring is important for the multivariate 4 generating function. The content should be replaced by color for d -complete posets. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. d -complete poset . . Contents of this section . . . The d -complete posets arise from the dominant 1 minuscule heaps of the Weyl groups of simply-laced Kac-Moody Lie algebras. . . Proctor gave completely combinatorial description of 2 d -complete poset, which is a graded poset with d -complete coloring. . . Proctor showed that any d -complete poset can be 3 obtained from the 15 irreducible classes by slant-sum . . . The d -complete coloring is important for the multivariate 4 generating function. The content should be replaced by color for d -complete posets. . . Okada defined ( q , t ) -weight W P ( π ; q , t ) for d-compete 5 posets. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Double-tailed diamond poset . . Definition . . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Double-tailed diamond poset . . Definition . . The double-tailed diamond poset d k ( 1 ) is the poset depicted below: top k − 2 side side k − 2 bottom . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Double-tailed diamond poset . . Definition . . The double-tailed diamond poset d k ( 1 ) is the poset depicted below: top k − 2 side side k − 2 bottom A d k -interval is an interval isomorphic to d k ( 1 ) . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Double-tailed diamond poset . . Definition . . The double-tailed diamond poset d k ( 1 ) is the poset depicted below: top k − 2 side side k − 2 bottom A d k -interval is an interval isomorphic to d k ( 1 ) . A d − k -interval ( k ≥ 4 ) is an interval isomorphic to d k ( 1 ) − { top } . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Double-tailed diamond poset . . Definition . . The double-tailed diamond poset d k ( 1 ) is the poset depicted below: top k − 2 side side k − 2 bottom A d k -interval is an interval isomorphic to d k ( 1 ) . A d − k -interval ( k ≥ 4 ) is an interval isomorphic to d k ( 1 ) − { top } . A d − 3 -interval consists of three elements x , y and w such that w is covered by x and y . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Definition of d -complete poset . . Definition . . A poset P is d -complete if it satisfies the following three conditions for every k ≥ 3: . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . Definition of d -complete poset . . Definition . . A poset P is d -complete if it satisfies the following three conditions for every k ≥ 3: . . If I is a d − k -interval, then there exists an element v such 1 that v covers the maximal elements of I and I ∪ { v } is a d k -interval. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . Definition of d -complete poset . . Definition . . A poset P is d -complete if it satisfies the following three conditions for every k ≥ 3: . . If I is a d − k -interval, then there exists an element v such 1 that v covers the maximal elements of I and I ∪ { v } is a d k -interval. . . If I = [ w , v ] is a d k -interval and the top v covers u in P , 2 then u ∈ I . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Definition of d -complete poset . . Definition . . A poset P is d -complete if it satisfies the following three conditions for every k ≥ 3: . . If I is a d − k -interval, then there exists an element v such 1 that v covers the maximal elements of I and I ∪ { v } is a d k -interval. . . If I = [ w , v ] is a d k -interval and the top v covers u in P , 2 then u ∈ I . . . There are no d − k -intervals which differ only in the minimal 3 elements. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Examples . rooted tree swivel shape shifted shape . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Properties of d -complete posets . . Fact . . If P is a connected d -complete poset, then . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Properties of d -complete posets . . Fact . . If P is a connected d -complete poset, then (a) P has a unique maximal element. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Properties of d -complete posets . . Fact . . If P is a connected d -complete poset, then (a) P has a unique maximal element. (b) P is ranked, i.e., there exists a rank function r : P → N such that r ( x ) = r ( y ) + 1 if x covers y . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Properties of d -complete posets . . Fact . . If P is a connected d -complete poset, then (a) P has a unique maximal element. (b) P is ranked, i.e., there exists a rank function r : P → N such that r ( x ) = r ( y ) + 1 if x covers y . . . . . Fact . . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Properties of d -complete posets . . Fact . . If P is a connected d -complete poset, then (a) P has a unique maximal element. (b) P is ranked, i.e., there exists a rank function r : P → N such that r ( x ) = r ( y ) + 1 if x covers y . . . . . Fact . . (a) Any connected d -complete poset is uniquely decomposed into a slant sum of one-element posets and slant-irreducible d -complete posets. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Properties of d -complete posets . . Fact . . If P is a connected d -complete poset, then (a) P has a unique maximal element. (b) P is ranked, i.e., there exists a rank function r : P → N such that r ( x ) = r ( y ) + 1 if x covers y . . . . . Fact . . (a) Any connected d -complete poset is uniquely decomposed into a slant sum of one-element posets and slant-irreducible d -complete posets. (b) Slant-irreducible d -complete posets are classified into 15 families : shapes, shifted shapes, birds, insets, tailed insets, banners, nooks, swivels, tailed swivels, tagged swivels, swivel shifts, pumps, tailed pumps, near bats, bat. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . . . . . . . . Irreducible d -complete poset . . Definition (Filter) . . Let S be a subset of a poset P . If S satisfies the condition x ∈ S and y ≥ x ⇒ y ∈ S then S is said to be a filter . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Irreducible d -complete poset . . Definition (Filter) . . Let S be a subset of a poset P . If S satisfies the condition x ∈ S and y ≥ x ⇒ y ∈ S then S is said to be a filter . . . . . Irreducible d -complete posets . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . Irreducible d -complete poset . . Definition (Filter) . . Let S be a subset of a poset P . If S satisfies the condition x ∈ S and y ≥ x ⇒ y ∈ S then S is said to be a filter . . . . . Irreducible d -complete posets . . Proctor defined the notion of irreducible d -complete 1 posets and classified them into 15 families. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . Irreducible d -complete poset . . Definition (Filter) . . Let S be a subset of a poset P . If S satisfies the condition x ∈ S and y ≥ x ⇒ y ∈ S then S is said to be a filter . . . . . Irreducible d -complete posets . . Proctor defined the notion of irreducible d -complete 1 posets and classified them into 15 families. . . A filter of a d -complete poset is a d -complete poset . 2 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Irreducible d -complete poset . . Definition (Filter) . . Let S be a subset of a poset P . If S satisfies the condition x ∈ S and y ≥ x ⇒ y ∈ S then S is said to be a filter . . . . . Irreducible d -complete posets . . Proctor defined the notion of irreducible d -complete 1 posets and classified them into 15 families. . . A filter of a d -complete poset is a d -complete poset . 2 . . 1) Shapes, 2) Shifted shapes, 3) Birds, 4) Insets, 5) Tailed 3 insets, 6) Banners, 7) Nooks, 8) Swivels, 9) Tailed swivels, 10) Tagged swivels, 11) Swivel shifteds, 12) Pumps, 13) Tailed pumps, 14) Near bats, 15) Bat . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Shapes . . Definition (Shapes) . . 1) Shapes . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Shifted shapes . . Definition (Shifted shapes) . . 2) Shifted shapes . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Birds . . Definition (Birds) . . 3) Birds . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Insets . . Definition (Insets) . . 4) Insets . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Tailed insets . . Definition (Tailed insets) . . 5) Tailed insets . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Banners . . Definition (Banners) . . 6) Banners . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Nooks . . Definition (Nooks) . . 7) Nooks . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Swivels . . Definition (Swivels) . . 8) Swivels . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Tailed swivels . . Definition (Tailed swivels) . . 9) Tailed swivels . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Tagged swivels . . Definition (Tagged swivels) . . 10) Tagged swivels . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Swivel shifteds . . Definition (Swivel shifteds) . . 11) Swivel shifteds . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Pumps . . Definition (Pumps) . . 12) Pumps . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Tailed pumps . . Definition (Tailed pumps) . . 13) Tailed pumps . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Near bats . . Definition (Near bats) . . 14) Near bats . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Bat . . Definition (Bat) . . 15) Bat . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Colored hook length property of d -complete posets . . Theorem (Peterson-Proctor) . . d -complete poset has the colored hook-length property. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Colored hook length property of d -complete posets . . Theorem (Peterson-Proctor) . . d -complete poset has the colored hook-length property. . . . . Remark . . Recently, Jan Soo Kim and Meesue Yoo gave a proof of the hook-length property by q -integral. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

Leaf Posets . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . Leaf Posets . . Contents of this section . . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . Leaf Posets . . Contents of this section . . . We define 6 family of posets, which we call the basic leaf 1 posets. (It is not possible to define “irreducibility”.) . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Leaf Posets . . Contents of this section . . . We define 6 family of posets, which we call the basic leaf 1 posets. (It is not possible to define “irreducibility”.) . . Leaf poset is defined as joint-sum of the basic leaf 2 posets. (“joint-sum” is a more genral notion than the slant-sum.) . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . Leaf Posets . . Contents of this section . . . We define 6 family of posets, which we call the basic leaf 1 posets. (It is not possible to define “irreducibility”.) . . Leaf poset is defined as joint-sum of the basic leaf 2 posets. (“joint-sum” is a more genral notion than the slant-sum.) . . Any d -complete poset is a leaf poset. 3 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . Leaf Posets . . Contents of this section . . . We define 6 family of posets, which we call the basic leaf 1 posets. (It is not possible to define “irreducibility”.) . . Leaf poset is defined as joint-sum of the basic leaf 2 posets. (“joint-sum” is a more genral notion than the slant-sum.) . . Any d -complete poset is a leaf poset. 3 . . If two posets has colored hook-length property then their 4 joint-sum has colored hook-length property. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Leaf Posets . . Contents of this section . . . We define 6 family of posets, which we call the basic leaf 1 posets. (It is not possible to define “irreducibility”.) . . Leaf poset is defined as joint-sum of the basic leaf 2 posets. (“joint-sum” is a more genral notion than the slant-sum.) . . Any d -complete poset is a leaf poset. 3 . . If two posets has colored hook-length property then their 4 joint-sum has colored hook-length property. . . The colored hook-length property of the basic leaf posets 5 . . . reduces to the Schur function identities. . . . . . . Masao Ishikawa Leaf poset and hook length property

蔦 笹 銀杏 樅 菊 藤 . Basic Leaf Posets . . Definition . . . . . . . . Masao Ishikawa Leaf poset and hook length property . . .

蔦 笹 樅 菊 藤 . Basic Leaf Posets . . Definition . . ginkgo( 銀杏 ) . . . . . . Masao Ishikawa Leaf poset and hook length property . . .

蔦 樅 菊 藤 . Basic Leaf Posets . . Definition . . bamboo( 笹 ) ginkgo( 銀杏 ) . . . . . . Masao Ishikawa Leaf poset and hook length property . . .

樅 菊 藤 . Basic Leaf Posets . . Definition . . ivy( 蔦 ) bamboo( 笹 ) ginkgo( 銀杏 ) . . . . . . Masao Ishikawa Leaf poset and hook length property . . .

樅 菊 . Basic Leaf Posets . . Definition . . ivy( 蔦 ) bamboo( 笹 ) ginkgo( 銀杏 ) wisteria( 藤 ) . . . . . . Masao Ishikawa Leaf poset and hook length property . . .

菊 . Basic Leaf Posets . . Definition . . ivy( 蔦 ) bamboo( 笹 ) ginkgo( 銀杏 ) fir( 樅 ) wisteria( 藤 ) . . . . . . Masao Ishikawa Leaf poset and hook length property . . .

. Basic Leaf Posets . . Definition . . ivy( 蔦 ) bamboo( 笹 ) ginkgo( 銀杏 ) fir( 樅 ) chrysanthemum( 菊 ) wisteria( 藤 ) . . . . . . Masao Ishikawa Leaf poset and hook length property basic leaf posets. . . .

. Definition . . (i) m ≥ 2, α = ( α 1 , α 2 , . . . , α m ) , β = ( β 1 , β 2 , . . . , β m ) : strict partitions γ α 1 α 2 α 3 c γ α m c γ c γ c γ = γ β 1 β 2 β 3 β m G ( α, β, γ ) := ginkgo ( 銀杏 ) . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Definition . . (ii) m ≥ 2, α = ( α 1 , α 2 , . . . , α m ) , β = ( β 1 , β 2 , . . . , β m − 1 ) , γ = ( γ 1 , γ 2 ) : strict partition, v = 1 , 2 β 1 α 1 α 2 α 3 γ 1 γ 2 α 4 c γ v α m c γ v B ( α, β, γ, v ) := c γ v β 1 β 2 β 3 β m − 1 bamboo ( 笹 ) . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Definition . . (iii) α = ( α 1 , α 2 , α 3 ) , β = ( β 1 , β 2 , β 3 , β 4 , β 5 ) , γ = ( γ 1 , γ 2 ) : β 1 α 1 strict partition for v = 1 , 2 α 2 α 3 γ 1 γ 2 γ 1 γ 2 α 1 α 2 α 3 c γ v c γ v I ( α, β, γ, v ) := β 1 β 2 β 3 β 4 β 5 ivy ( 蔦 ) . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Definition . . (iv) m ≥ 2, α = ( α 1 , α 2 , . . . , α m ) , β = ( β 1 , β 2 ) , γ = ( γ 1 , γ 2 ) : strict partition γ 1 α 1 α 2 α 3 α 4 W ( α, β, γ, v ) = α 5 β 1 β 2 γ 1 γ 2 α m β 1 β 2 c h v γ 1 γ 2 ( g 1 , g 2 , h v ) g 1 g 2 wisteria ( 藤 ).  ( β 1 , β 2 , γ v ) if m : even   :=   ( γ 1 , γ 2 , β v ) if m : odd   . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Definition β 1 α 1 . . α 2 (v) m ≥ 3, α 3 γ 1 α = ( α 1 , α 2 , α 3 ) , γ 2 α s β = ( β 1 , β 2 , . . . , β m − 1 ) , α t γ 1 γ 2 γ 1 γ 2 γ = ( γ 1 , γ 2 ) α s α t : strict partitions, γ 1 γ 2 s , t ≥ 1 (1 ≤ s < t ≤ 3), g 1 g 2  s or t if m : even ,   c hv v =   1 or 2 if m : odd  β 1 β 2 β 3 β 4 β 5 β 6 β 7 β m − 1  F ( α, β, γ, s , t , v ) = fir ( 樅 ).  ( β 1 , β 2 , α v ) if m : even   ( g 1 , g 2 , h v ) :=   ( α s , α t , γ v ) if m : odd   . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Definition . . (vi) α = ( α 1 , α 2 , α 3 ) , β = ( β 1 , β 2 , β 3 , β 4 ) and γ = ( γ 1 , γ 2 ) : strict partitions, δ ≥ 0 for v = 1 , 2 , 3 , 4 β 1 α 1 α 2 α 3 γ 1 γ 2 γ 1 γ 2 α 1 α 2 α 3 C ( α, β, γ, v ) = β 1 β 2 β 3 β 4 c β v γ 1 γ 2 chrysanthemum ( 菊 ). . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . . . . . . . . . . . . . . . . . . Goal of This Talk . . Property of leaf posets . . . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . . . . . Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . . . . Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . 1) Shapes, 3) Birds ⊆ Ginkgo 1 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . . . Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . 1) Shapes, 3) Birds ⊆ Ginkgo 1 . . 2) Shifted shapes, 6) Banners ⊆ Wisteria 2 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . . Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . 1) Shapes, 3) Birds ⊆ Ginkgo 1 . . 2) Shifted shapes, 6) Banners ⊆ Wisteria 2 . . 5) Tailed insets, 4) Insets ⊆ Bamboo 3 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . 1) Shapes, 3) Birds ⊆ Ginkgo 1 . . 2) Shifted shapes, 6) Banners ⊆ Wisteria 2 . . 5) Tailed insets, 4) Insets ⊆ Bamboo 3 . . 7) Nooks, 9) Tailed swivels, 10) Tagged swivels, 11) Swivel 4 shifteds ⊆ Fir . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . 1) Shapes, 3) Birds ⊆ Ginkgo 1 . . 2) Shifted shapes, 6) Banners ⊆ Wisteria 2 . . 5) Tailed insets, 4) Insets ⊆ Bamboo 3 . . 7) Nooks, 9) Tailed swivels, 10) Tagged swivels, 11) Swivel 4 shifteds ⊆ Fir . . 8) Swivels ⊆ Ivy 5 . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . 1) Shapes, 3) Birds ⊆ Ginkgo 1 . . 2) Shifted shapes, 6) Banners ⊆ Wisteria 2 . . 5) Tailed insets, 4) Insets ⊆ Bamboo 3 . . 7) Nooks, 9) Tailed swivels, 10) Tagged swivels, 11) Swivel 4 shifteds ⊆ Fir . . 8) Swivels ⊆ Ivy 5 . . 12) Pumps, 13) Tailed pumps, 14) Near bats, 15) Bat ⊆ 6 Chrysanthemum . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Goal of This Talk . . Property of leaf posets . . . Any d -complete poset is a leaf poset. 1 . . 1) Shapes, 3) Birds ⊆ Ginkgo 1 . . 2) Shifted shapes, 6) Banners ⊆ Wisteria 2 . . 5) Tailed insets, 4) Insets ⊆ Bamboo 3 . . 7) Nooks, 9) Tailed swivels, 10) Tagged swivels, 11) Swivel 4 shifteds ⊆ Fir . . 8) Swivels ⊆ Ivy 5 . . 12) Pumps, 13) Tailed pumps, 14) Near bats, 15) Bat ⊆ 6 Chrysanthemum . . . . Theorem . . A leaf poset has multi-colored hook length property. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. Schur Function . . Definition (Schur Function) . . If λ = ( λ 1 , . . . , λ n ) is a partition of length ≤ n , then � x λ 1 + n − 1 x λ n � . . . � � � � 1 1 � � . . ... � � . . � � . . � � � � � � x λ 1 + n − 1 x λ n . . . � � � � n n s λ ( x 1 , . . . , x n ) = . x n − 1 � � . . . 1 � � � 1 � � . . � ... � . . � � . . � � � � � x n − 1 � � . . . 1 � � n The Schur functions are the irreducible characters of the polynomial representations of the General Linear Group. . . . . . . . . . Masao Ishikawa Leaf poset and hook length property

. . . . . . . . . . Symmetric Functions . . Theorem (Cauchy’s formula) . If n is a positive integer, then n n 1 ∑ ∏ ∏ s λ ( x 1 , . . . , x n ) s λ ( y 1 , . . . , y n ) = . 1 − x i y j i = 1 j = 1 . λ . . . . . . Masao Ishikawa Leaf poset and hook length property

. Symmetric Functions . . Theorem (Cauchy’s formula) . If n is a positive integer, then n n 1 ∑ ∏ ∏ s λ ( x 1 , . . . , x n ) s λ ( y 1 , . . . , y n ) = . 1 − x i y j i = 1 j = 1 . λ . Proposition . If n is a positive integer, then n 1 ∏ ∑ h r ( x 1 , . . . , x n ) t n , = 1 − tx i j = 1 r ≥ 0 n n ∏ ∑ e r ( x 1 , . . . , x n ) t n ( 1 + tx i ) = j = 1 r = 0 where h r is the complete symmetric function and e r is the elementary symmetric function. . . . . . . . Masao Ishikawa Leaf poset and hook length property