Littlewood-Richardson coefficients, the hive model and Horn inequalities

Transcription

1 Littlewood-Richardson coefficients, the hive model and Horn inequalities Ronald C King School of Mathematics, University of Southampton Southampton, SO7 BJ, England Presented at: SLC 6 Satellite Seminar, University of Coimbra Coimbra, Portugal: September 28 Seoul - 28 p. /86

2 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 2/86

3 Key sources Littlewood-Richardson coefficients D.E. Littlewood and A. Richardson, Group characters and algebra, Phil. Trans. Roy. Soc. London A 233 (934) 99 4 D.E. Littlewood, Theory of Group Characters, 2nd Ed. Oxford: Clarendon Press, 94 The hive model A. Knutson and T. Tao, The honeycomb model of GL n (C) tensor products. I: Proof of the saturation conjecture, J. Amer. Math. Soc. 2 (999) 55 9 A.S. Buch, The saturation conjecture (after A. Knutson and T. Tao). With an appendix by W. Fulton, Enseign. Math. 46 (2) 43 6 Seoul - 28 p. 3/86

4 Key sources Horn inequalities A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 2 (962) Puzzles A. Knutson, T. Tao and C. Woodward, The honeycomb model of GL n (C) tensor products. II: Puzzles determine facets of the Littlewood-Richardson cone, Amer. Math. Soc. 7 (24) 9 48 V.I. Danilov and G.A. Koshevoy, Diskretnaya vognutost i ermitovy matritsy, Trudy Math. Inst. Steklova 24 (23) 68 9, in Russian. Title translation: Discrete convexity and Hermitian matrices Seoul - 28 p. 4/86

5 Key sources Polynomial property of stretched LR-coefficients E. Ehrhart, Sur les polyedres rationnels homothetiques a n dimensions, C. R. Acad. Sci. Paris Ser A 254 (962) H. Derksen and J. Weyman, On the Littlewood-Richardson polynomials, J. Algebra. 255 (22) E. Rassart, A polynomial property of Littlewood-Richardson coefficients, J Comb. Theo. Ser. A 7 (24) 6 79 Overview W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. 37 (2) Seoul - 28 p. 5/86

6 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 6/86

7 Schur functions Let n be a fixed positive integer and x = (x,x 2,...,x n ) a sequence of indeterminates. Let λ = (λ,λ 2,...,λ n ) be a partition of weight λ and length l(λ) n, so that λ λ 2 λ n. Then the Schur function s λ (x) is defined by: s λ (x) = x n+λ j j i i,j n x n j i. i,j n The Schur functions form a Z-basis of Λ n, polynomial symmetric functions of x,...,x n. the ring of Each Schur function s λ (x) may be interpreted as the character ch V λ (x) of an irrep of gl(n). Seoul - 28 p. 7/86

8 LR-coefficients Any product of Schur functions can be expressed as a linear sum of Schur functions: s λ (x) s µ (x) = ν c ν λµ s ν (x) The coefficients c ν λµ are the multiplicities appearing in the decomposition of the tensor product of irreps of gl(n): V λ V µ = ν c λ λµ V ν Each Littlewood-Richardson coefficient c ν λµ is a non-negative integer that may be evaluated by means of the Littlewood-Richardson rule Seoul - 28 p. 8/86

9 Young diagrams Each partition λ specifies a Young diagram F λ consisting of λ boxes arranged in l(λ) left adjusted rows of lengths λ λ 2... λ l(λ) >. The partition λ conjugate to λ is such that F λ is obtained from F λ by interchanging rows and columns. Ex: If λ = (4, 3, ) then λ = 8, l(λ) = 3, λ = (3, 2, 2, ), with F λ = F λ = Seoul - 28 p. 9/86

10 Skew Young diagrams Given partitions λ and ν such that all boxes of F λ are contained in F ν we write λ ν. Removing the boxes of F λ from F ν leaves the skew Young diagram F ν/λ Ex: If ν = (5, 4, 2) and λ = (3, ) then F ν/λ = The corresponding skew Schur function is such that s ν/λ (x) = µ c ν λµ s µ (x) Seoul - 28 p. /86

11 Littlewood-Richardson rule Fill the boxes of the Young diagram F λ with s Then fill the boxes of the skew Young diagram F ν/λ with µ i entries i for i =, 2,...,n. c ν λµ is the number of such diagrams with entries weakly increasing across rows from left to right strictly increasing down columns from top to bottom satisfying the lattice permutation rule - i.e. at every stage in the sequence of non-zero entries read from right to left across rows taken in turn from top to bottom # s #2 s #n s Seoul - 28 p. /86

12 Hence c ν λµ = 3. Seoul - 28 p. 2/86 Application of the LR-rule Ex: n = 4, λ = (4, 2), µ = (4, 3, 2), ν = (6, 5, 3, ) The only valid LR-diagrams Some invalid diagrams

13 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 3/86

14 Integer hives Knutson & Tao [99], as described by Buch [] An integer n-hive is a triangular graph with vertex labels a ij Z for i,j,i + j n. Ex: n = 4 a 4 a 3 a 3 a 2 a 2 a 22 a a a 2 a 3 a a a 2 a 3 a 4 Vertex labels increase along each edge from left to right Seoul - 28 p. 4/86

15 Relation between vertex and edge labels Edge labels are the non-negative differences between neighbouring vertex labels Vertex and edge labels for the two types of elementary triangle q p ρ r σ τ σ τ p ρ r q In each case σ = q p, τ = r q, ρ = r p so that automatically we have σ + τ = ρ in any such triangle Seoul - 28 p. 5/86

16 Hive conditions Three distinct types of rhombi with vertex labels: a b d a b b c a c d c d The hive condition for each rhombus: b + c a + d Three distinct types of rhombi with edge labels: α β δ γ δ γ α β β α γ δ The hive condition for each rhombus: α γ and β δ Note: The triangle edge condition implies α + δ = β + γ. Seoul - 28 p. 6/86

17 LR-hives vertex labels Definition An LR-hive is an integer n-hive for which all rhombi satisfy the hive conditions; boundaries determined by partitions λ, µ, ν with l(λ),l(µ),l(ν) n and λ + µ = ν ; boundary vertex labels as shown: λ λ + µ λ + λ 2 λ + µ + µ 2 λ ν ν + ν 2 ν = λ + µ Seoul - 28 p. 7/86

18 LR-hives edge labels Definition An LR-hive is an integer n-hive for which all rhombi satisfy the hive conditions; boundaries determined by partitions λ, µ, ν with l(λ),l(µ),l(ν) n and λ + µ = ν ; boundary edge labels as shown: λ n µ µ 2 λ 2 λ µ n ν ν 2 ν n Seoul - 28 p. 8/86

19 Bijection between LR-diagrams and LR-hives Lemma A bijection between LR integer n-hives, H, and LR-diagrams, D, is provided by the formula: a ij = # of entries i in the first i + j rows of D for the (i, j)th vertex label in H, for all i, j such that i,j,i+j n. Theorem The LR-coefficient c ν λµ is the number of LR-hives with boundary labels determined by λ, µ and ν. Note: Neither this theorem nor the Littlewood-Richardson rule allows us to see whether or not a given LR-coefficient c ν λµ is non-zero. Seoul - 28 p. 9/86

20 Example of bijection Ex: n = 4, λ = (753), µ = (742), ν = (9964) a ij : Seoul - 28 p. 2/86

21 Example of bijection Ex: n = 4, λ = (753), µ = (742), ν = (9964) a ij : Note The given edge labels are sufficient to fix all edge labels using the triangle condition Seoul - 28 p. 2/86

22 LR-hives showing that c ,742 = Seoul - 28 p. 22/86

23 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 23/86

24 Non-zero conditions We know that c ν λµ is the number of LR-hives with boundary labels determined by λ, µ and ν For given λ, µ and ν we would like some way of determining if c ν λµ is non-zero Horn [62] defined a set of inequalities and conjectured that they gave necessary and sufficient conditions for the solution of a problem later realised to be equivalent to that of LR-coefficients being non-zero The validity of Horn s conjecture was proved by the efforts of Klyachko [98], Knutson and Tao [99], Belkale [], and Knutson, Tao and Woodward [4] For a comprehensive review see Fulton [] Seoul - 28 p. 24/86

25 Partial sums Let N = {, 2,...,n}, then for fixed r, with r n, let I = {i,i 2,...,i r } N and I = N\I. For any partition λ let: ps (λ) I = λ i + λ i2 + + λ ir. If i < i 2 < < i r then let part (I) = (i r r,...,i 2 2,i ). Let T n r be the set of triples (I,J,K) with I,J,K N and part (K) #I = #J = #K = r with cpart (I)part (J) >. Let R n r be the set of triples (I,J,K) with I,J,K N and part (K) #I = #J = #K = r with cpart (I)part (J) =. Seoul - 28 p. 25/86

26 Non-zero conditions Theorem: The LR-coefficient c ν λµ is non-zero if and only if ν = λ + µ and Horn s inequalities, ps (ν) K ps (λ) I + ps (µ) J, are satisfied for all r =, 2,...,n and all (I,J,K) T n r Note: Not all of Horn s inequalities are essential. Horn s essential inequalities are those for which (I,J,K) R n r - but where do they come from? Seoul - 28 p. 26/86

27 Puzzles - Knutson, Tao & Woodward [4] Definition A puzzle is a diagram on a triangular lattice in which edges are distinguished so that it is composed of copies of the following pieces oriented in any way so as to fit: Seoul - 28 p. 27/86

28 Puzzles Definition A puzzle is a diagram on a triangular lattice in which edges are distinguished so that it is composed of copies of the following pieces oriented in any way so as to fit: Seoul - 28 p. 28/86

29 Puzzles Definition A puzzle is a diagram on a triangular lattice in which edges are distinguished so that it is composed of copies of the following pieces oriented in any way so as to fit: Seoul - 28 p. 29/86

30 Hive plan or labyrinth Definition A hive plan is made up of corridors, dark rooms and light rooms obtained by deleting interior edges of a puzzle: Seoul - 28 p. 3/86

31 Hive plan or labyrinth Definition A hive plan is made up of shaded corridors, dark rooms and light rooms obtained by deleting interior edges of a puzzle: Seoul - 28 p. 3/86

32 Hive plan or labyrinth Definition A hive plan is made up of corridors, blue rooms and red rooms obtained by deleting interior edges of a puzzle: Seoul - 28 p. 32/86

33 Link between puzzles and Horn triples (I,J,K) is Horn triple if it specifies the positions of the thick edges on the boundary of any puzzle. It is essential if the puzzle with these boundary thick edges is unique. For I = (, 2, 4), J = (2, 3, 4) and K = (2, 3, 5) we have: Seoul - 28 p. 33/86

34 Each Horn triple defines an inequality λ 5 µ γ 3 λ 4 µ 2 α 4 λ 3 γ γ 2 α 3 β 3 µ 3 λ 2 β 4 µ 4 λ α 2 α β 2 α 5 β 5 β γ 4 µ 5 ν ν 2 ν 3 ν 4 ν 5 ν 2 + ν 3 + ν 5 (ν 2 + ν 3 ) + γ 4 = (α + α 2 + β + β 2 ) + γ 4 λ + α 2 + β + β 2 + γ 4 λ + λ 2 + β + β 2 + γ 4 λ + λ 2 + β 3 + β 2 + γ 4 λ + λ 2 + (β 3 + β 4 + γ 4 ) = λ + λ 2 + (α 4 + µ 2 + µ 3 + µ 4 ) λ + λ 2 + λ 4 + µ 2 + µ 3 + µ 4 That is ps (ν) K ps (λ) I + ps (µ) J. Seoul - 28 p. 34/86

35 Complementary Horn inequalities λ 5 µ γ 3 λ 4 µ 2 α 4 λ 3 γ γ 2 α 3 β 3 µ 3 λ 2 β 4 µ 4 λ α 2 α β 2 α 5 β 5 β γ 4 µ 5 ν ν 2 ν 3 ν 4 ν 5 The same procedure applied to thin-edge inequalities gives ν + ν 4 γ + ν 4 = γ + (α 5 + β 5 ) γ + α 3 + β 5 (γ + α 3 ) + µ 5 = (λ 3 + γ 2 ) + µ 5 λ 3 + γ 3 + µ 5 = λ 3 + λ 5 + µ + µ 5 That is ps (ν) K ps (λ) I + ps (µ) J Seoul - 28 p. 35/86

36 Significance of inequalities derived from puzzles Each inequality ps (ν) K ps (λ) I + ps (µ) J derived from a puzzle must be satisfied if c ν λµ is to be non-zero To show that a puzzle triple (I,J,K) is a Horn triple, and the inequality a Horn inequality, we must make a connection between puzzles with thick boundary edges specified by (I,J,K) and c part (K) part (I),part (J) First note that the passage from I to part (I) is easy Ex If I = {, 3, 6, 8} then (, 3, 6, 8) (, 2, 3, 4) = (,, 3, 4) so that part (I) = (4, 3, ) Seoul - 28 p. 36/86

37 How may puzzles are there? Theorem The number of puzzles with the positions of the thick edges on the boundary specified by (I,J,K) is given by c part (K) part (I),part (J) Ex: n = 5, r = 2, I = (, 2, 4), J = (2, 3, 4), K = (2, 3, 5) In this case part (K) cpart (I),part (J) = c3, = and there exists just the one puzzle identified earlier Seoul - 28 p. 37/86

38 A map from puzzles to hives Let the thick edges of the puzzle be specified by (I,J,K) For M = I,J,K in turn, label each thick boundary edge of the puzzle by the corresponding row part of part (M) Scale the length of all thin edges by t and let t Seoul - 28 p. 38/86

39 A map from puzzles to hives contd In each thick edged room set all parallel edge labels equal using the triangle condition wherever required Reflect the resulting diagram in its vertical axis of symmetry to obtain a hive Lemma This map provides a bijection between puzzles with thick edges specified by (I, J, K) and LR-hives with boundary specified by part (J), part (I), part (K) Seoul - 28 p. 39/86

40 Corollaries Each triple (I,J,K) is a Horn triple if and only if it specifies the thick boundary edges of a puzzle The corresponding inequality ps (ν) K ps (λ) I + ps (µ) J defined by the puzzle is a Horn inequality The number of puzzles specified by (I,J,K) is equal to c part (K) part (J),part (I) = cpart (K) part (I),part (J) The puzzle is said to be rigid if it is unique, that is c part (K) part (J),part (I) inequality is essential =, and the corresponding Horn The LR-coefficient c ν λµ is non-zero if and only if every essential Horn inequality is satisfied Seoul - 28 p. 4/86

41 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 4/86

42 Consequences of any Horn equality λ 5 µ γ 3 λ 4 µ 2 α 4 λ 3 γ γ 2 α 3 β 3 µ 3 λ 2 β 4 µ 4 λ α 2 α β 2 α 5 β 5 β γ 4 µ 5 ν ν 2 ν 3 ν 4 ν 5 All sequences of inequalities become equalities. ν 2 + ν 3 + ν 5 = (ν 2 + ν 3 ) + γ 4 = (α + α 2 + β + β 2 ) + γ 4 = λ + α 2 + β + β 2 + γ 4 = λ + λ 2 + β + β 2 + γ 4 = λ + λ 2 + β 3 + β 2 + γ 4 = λ + λ 2 + (β 3 + β 4 + γ 4 ) = λ +λ 2 +(α 4 +µ 2 +µ 3 +µ 4 ) = λ +λ 2 +λ 4 +µ 2 +µ 3 +µ 4. Seoul - 28 p. 42/86

43 Edge label equalities λ 5 µ γ 3 λ 4 µ 2 α 4 λ 3 γ γ 2 α 3 β 3 µ 3 λ 2 β 4 µ 4 λ α 2 α β 2 α 5 β 5 β γ 4 µ 5 ν ν 2 ν 3 ν 4 ν 5 ν + ν 4 = γ + ν 4 = γ + (α 5 + β 5 ) = γ + α 3 + β 5 = (γ + α 3 ) + µ 5 = (λ 3 + γ 2 ) + µ 5 = λ 3 + γ 3 + µ 5 = λ 3 + λ 5 + µ + µ 5 These equalities imply: ν 5 = γ 4, α = λ, α 2 = λ 2, β = β 3, β 2 = β 4, α 4 = λ 4, β 5 = µ 5, ν = γ, γ 2 = γ 3. Seoul - 28 p. 43/86

44 Factorisation of LR-hives λ 5 µ γ 2 λ 4 µ 2 λ 4 λ 3 ν γ 2 α 3 β µ 3 λ 2 β 2 µ 4 ν 5 λ λ 2 λ β 2 α 3 µ 5 β µ 5 ν ν 2 ν 3 ν 4 ν 5 λ 4 µ 2 λ 5 γ 2 µ λ 2 β µ 3 λ 3 α 3 µ 5 λ β 2 µ 4 ν ν 2 ν 3 ν 5 ν 4 Seoul - 28 p. 44/86

45 Illustration of H n and subhives H r,h n r λ 5 µ λ 4 µ 2 λ 3 µ 3 λ 2 µ 4 λ µ 5 ν ν 2 ν 3 ν 4 ν 5 λ λ 2 λ 4 µ 2 µ 3 µ 4 λ 3 λ 5 µ µ 5 ν ν 2 ν 3 ν 5 ν 4 Seoul - 28 p. 45/86

46 LR-coefficient factorisation Lemma In the case of any Horn equality and a corresponding puzzle, the deletion of redundant corridors from any LR-hive H n gives a pair of LR-subhives H r and H n r. Lemma In the case of any essential Horn equality, this map from the LR-hives H n to pairs of LR-hives H r and H n r is a bijection. Theorem then c ν λµ If an essential Horn inequality is saturated factorises. Definition If all essential Horn inequalities are strict c ν λµ is said to be primitive. Seoul - 28 p. 46/86

47 LR factorisation example Ex: n = 5, r = 3, n r = 2: Hence λ = (9, 7, 6, 2, ), µ = (3, 5, 3,, ), ν = (4, 2,, 5, 4). I = {, 2, 4}, J = {2, 3, 4}, K = {2, 3, 5}. λ I = (9, 7, 2), µ J = (5, 3, ), ν K = (2,, 4) λ I = (6, ), µ J = (3, ), ν K = (4, 5) ps (ν) K = 27 = = ps (λ) I + ps (µ) J c (4,2,,5,4) (9,7,6,2,),(3,5,3,,) = c(2,,4) (9,7,2),(5,3,) c(4,5) (6,),(3,) = 2 = 2 Note: This is an example of the reduction of an LR-coefficient, since c (4,2,,5,4) (9,7,6,2,),(3,5,3,,) = c(2,,4) (9,7,2),(5,3,) = 2 Seoul - 28 p. 47/86

48 Proof of factorisation To be shown: the corridors R n of H n are redundant; the dark rooms constitute an LR-hive H r ; the light rooms constitute an LR-hive H n r ; any LR hives H r,h n r joined by R n gives an LR-hive H n. To be checked that the Horn equality implies: all corridor edge labels fixed; LR hive conditions for any rhombus split by corridor; LR hive conditions across corridor/dark room boundary; LR hive conditions across corridor/light room boundary. Seoul - 28 p. 48/86

49 Corridor edges fixed Hive conditions: γ σ τ α, γ 2 σ 2 τ 2 α 2. Horn equality: γ + γ 2 = α + α 2. Implies: γ = α and γ 2 = α 2. Implies: γ = σ = τ = α and γ 2 = σ 2 = τ 2 = α 2. α α τ α 2 α α 2 σ τ 2 α α 2 γ σ 2 α α 2 γ 2 α 2 Seoul - 28 p. 49/86

50 Deletion of corridor Initial hive conditions: γ σ, σ τ, τ α. Implies final hive condition: γ α. α β ρ τ ρ β α ρ δ σ ρ δ γ γ Horn equality α β = ρ = γ δ implies α + δ = β + γ. Seoul - 28 p. 5/86

51 Paths - gentle and good Path: a continuous sequence of connected corridor walls with dark rooms, thick-edged -regions, on the right and light rooms, thin-edged -regions, on the left. Gentle path: at each vertex the deviation is or ±π/3. Gentle loop: a gentle path that forms a closed interior loop. An edge is good if it forms the short diagonal of a rhombus satisfying the hive condition, otherwise it is bad. Good path: gentle path along which all the edges are good, ie with the hive condition satisfied across each edge. Seoul - 28 p. 5/86

52 Good paths and factorisation Observations When subdividing two LR subhives, H r and H n r, and inserting corridors to create a hive H n, this hive will be an LR hive if and only if each edge of every internal corridor wall of the corresponding hive plan is good. Let the boundary labels λ,µ,ν of the hive H n be such that for a given puzzle specified by (I,J,K) the corresponding Horn inequality is saturated, ie. ps (ν) K = ps (λ) I + ps (µ) J. If in the corresponding hive plan each edge of every internal corridor wall lies on a good path, then the LR-coefficient c ν λµ factorises. Seoul - 28 p. 52/86

53 Start of good paths Lemma The first edge of any path starting from any boundary is good Each boundary has edges specified by a partition: α γ. Horn equality applied to corridors: β = α or β = γ. Hence β = α γ or β = γ α so that in both cases OP is good. α β P O γ β α O γ P Seoul - 28 p. 53/86

54 Path along an interior corridor wall Initial hive conditions: α β Horn equality applied to corridors: δ = γ. PO good γ α δ = γ α β OR good OR bad δ > β γ = δ > β α PO bad R β α β α O P R O δ γ δ γ P R β α O P R β α O δ γ δ γ P Seoul - 28 p. 54/86

55 Path reaching a vertex Initial hive condition: α β; Horn equality δ = β PO good γ α γ α β = δ OR good OR bad γ > δ γ > δ = β α PO bad β R O α P β δ δ γ γ R O α P β δ R O α γ P β δ R O α γ P Seoul - 28 p. 55/86

56 Intersecting paths Assume PO good, so that β α. Horn equality γ = δ QO good γ β δ = γ β α OS good OS bad δ < α γ = γ = δ < α β QO bad S S δ R S δ R O β Q α β P δ R O δ γ γ β O α Q P S R O Q α γ γ β α Q P P Seoul - 28 p. 56/86

57 Union of good paths Good paths cover all interior corridor walls All LR hive conditions satisfied Hence we have factorisation Seoul - 28 p. 57/86

58 Bad edges and gentle loops Good paths may not cover all interior corridor edges If there exists a bad edge then its predecessor on some gentle path must also be bad A reverse gentle path of bad edges cannot reach the boundary, since all gentle paths start from the boundary with a good edge A reverse gentle path of bad edges must therefore continue indefinitely Since there are only a finite number of edges, it follows that there may only be bad edges if there exists a gentle loop Seoul - 28 p. 58/86

59 Example exhibiting a gentle loop n =, r = 5. I = (, 2, 4, 6, 8),J = (, 3, 4, 7, 9),K = (2, 4, 6, 8, ). Seoul - 28 p. 59/86

60 Example of a gentle loop Gentle loop is any closed interior gentle path. Seoul - 28 p. 6/86

61 Good paths Good paths do not include all interior corridor walls. Seoul - 28 p. 6/86

62 Obstruction to good paths Good paths do not include all interior corridor walls. Seoul - 28 p. 62/86

63 Example exhibiting a gentle loop Ex: n = 6, r = 3, I = J = (, 3, 5) and K = (2, 4, 6). There exist two puzzles, each exhibiting a gentle loop P: P2: The corresponding inessential saturated Horn inequality ν 2 + ν 4 + ν 6 = λ + λ 3 + λ 5 + µ + µ 3 + µ 5 is satisfied by λ = µ = (22) and ν = (3322) There can be no corresponding factorisation since c ν λµ = c ,22 = = c 32 2,2 c 32 2,2 = c ν K λi µ J c ν K λi µ J Seoul - 28 p. 63/86

64 Rigid puzzles and gentle loops Theorem [KTW 4] The hive plan of a puzzle has no gentle loops if only if the puzzle is rigid A puzzle is rigid if and only if the corresponding Horn inequality is essential All gentle paths in the hive plan of a rigid puzzle are good, ie have no bad edges This implies that in the case of any essential Horn equality, the map from the LR-hives H n to pairs of LR-hives H r and H n r is a bijection Hence we have proved: [KTT 8] Theorem If an essential Horn inequality is saturated then c ν λµ factorises. Seoul - 28 p. 64/86

65 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 65/86

66 Some relations between LR-coefficients Symmetries Inequalities c ν µ λ = cν λ µ c ν λ µ = c ν λ µ c µ ν λ = cν λ µ where λ is the conjugate of λ and λ = (m λ n,...,m λ 2,m λ ) c ν+(c ) λ+( a ), µ+( b ) cν λ µ c ν (c) λ (a), µ (b) cν λ µ for all non-negative integers a,b,c, with c = a + b where λ + ( a ) = (λ +,...,λ a +,λ a+,...,λ n ) and λ (a) = (λ,...,λ k,a,λ k+,...,λ n ) with λ k a > λ k+ Seoul - 28 p. 66/86

67 Hive based proofs of the symmetry relations Commutativity c ν µ λ = cν λ µ Proof: It remains an open problem to find a hive-based proof. We would like a bijection between the LR-hives for c ν µ λ and those for c ν λ µ The result is by no means obvious from the Littlewood-Richardson rule. Considerable effort has gone into devising combinatorial proofs See for example the work of Benkart, Sotille & Stroomer [96] and of Azenhas [99] Seoul - 28 p. 67/86

68 Hive based proofs of the symmetry relations Conjugacy c ν λ µ = c ν λ µ Let N = {, 2,...,n} = M M with M M = Note that if ζ = part (M) then ζ = part (M n ) where M n = {n + i i M} Ex: If n = and M = {, 3, 6, 8} then part (M) = (4, 3, ) while M = {2, 4, 5, 7, 9, } and M n = {, 2, 4, 6, 7, 9} so that part (M) = (3, 2, 2, ). Diagrammatically: Seoul - 28 p. 68/86

69 Proof of conjugacy contd. Recall that the number of n-puzzles with thick edges specified by I,J,K is c part (K) part (J),part (I) = cpart (K) part (I),part (J) This was proved by scaling all thick edges by t and allowing t Seoul - 28 p. 69/86

70 Proof of conjugacy contd. It remains to show that the number of n-puzzles with thin edges specified by I,J,K is c part (K n) part (I n ),part (J n ) This is proved by scaling all thin edges by t and allowing t Seoul - 28 p. 7/86

71 Hive based proofs of the symmetry relations Complementarity c µ ν λ = cν λ µ Proof Consider any LR-hive for c ν λ µ. Then For all edges parallel to the NE,SE,WE boundaries replace the edge labels α,β,γ by α,m β,m γ This breaks the triangle and the rhombus hive conditions and the result is not an LR-hive R R λ n µ λ n m µ µ 2 m µ 2 λ 2 λ 2 λ P ν ν 2 ν n µ n Q λ P m ν m ν 2 m ν n m µ n Q Seoul - 28 p. 7/86

72 Complementarity contd. Rotate clockwise through π/3 The result is an LR-hive for c µ ν λ R P λ n m µ m ν λ m µ 2 m ν 2 λ 2 λ 2 λ P m ν m ν 2 m ν n m µ n Q m ν n Q m µ n m µ 2 m µ λ n R Lemma The above sequence of two maps provides a bijection between the LR-hives of c ν λ µ and those of c µ ν λ Seoul - 28 p. 72/86

73 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 73/86

74 Column insertion Inequality c ν+(c ) λ+( a ), µ+( b ) cν λ µ for all a,b,c, with c = a + b Proof Since c c = if c = a + b, there exist a unique a, b LR-hive: Here each red, magenta and blue edge is labelled and all other edges are labelled Seoul - 28 p. 74/86

75 Column insertion The elementary rhombi take one or other of the following forms However, we can now interpret our multi-coloured LR-hive as being an LR-hive for c ν λ µ whose red, magenta and blue edge labels have all been increased by, with no change to the uncoloured edge labels Seoul - 28 p. 75/86

76 Column insertion This yields an LR-five for c ν+(c ) as can be seen by λ+( a ), µ+( b ) checking the hive conditions in each of the elementary rhombi Hence the number of c ν+(c ) LR-hives is at least as λ+( a ), µ+( b ) many as the number of c ν λ µ LR-hives Seoul - 28 p. 76/86

77 Row insertion Inequality c ν (c) λ (a), µ (b) cν λ µ with c = a + b Each LR-hive for c ν λ µ may be divided into regions as shown on the left below Redundant corridors are then inserted as shown on the right to give an LR-hive for c ν (c) λ (a), µ (b) b a A < a S Z T B < b = a A A S Z T B b B c C < c C c C Seoul - 28 p. 77/86

78 Row insertion b < a B a A S Z T B < b = a A A S Z T b B c C < c C c C The red line divides the hive into those regions for which NW edge labels are either a or > a The blue line divides the hive into those regions for which SW edge labels are either b or > b The cyan line divides the hive into those regions for which WE edge labels are either c or > c Seoul - 28 p. 78/86

79 Row insertion b < a B a A S Z T B < b = a A A S Z T b B c C < c C c C These lines always intersect at a single point Z Edges labelled a, b, c are inserted on the boundary Each edge along the first part of the red, blue and cyan lines is replaced by an appropriate redundant rhombus The point Z is replaced by a triangle with edge labels a, b, c, and all triangle and rhombus hive conditions are satisfied Seoul - 28 p. 79/86

80 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 8/86

81 Stretched LR coefficients Littlewood-Richardson coefficient c ν λµ Partition λ = (λ,λ 2,...,λ n ) stretching parameter t N Stretched partition tλ = (tλ,tλ 2,...,tλ n ) Stretched Littlewood-Richardson coefficient c tν tλ,tµ Ex: n = 3, λ = (2,, ), µ = (3, 2, ), ν = (4, 3, ) t = : c 43 2,32 = 2 t = 2: c ,64 = t = 3: c63,94 = 4 suggests c tν tλ,tµ = t +. Seoul - 28 p. 8/86

82 LR coefficients and polynomials Ex: Let c ν 42,532 = c and c tν t(42),t(532) = P(t). c = ν = (953) P(t) = c = 2 ν = (943) P(t) = (t + ) c = 3 ν = (844) P(t) = (t + )(t + 2)/2 c = 4 ν = (853) P(t) = (t + )(t + 2)(t + 3)/6 c = 4 ν = (7442) P(t) = (t + ) 2 c = 5 ν = (754) P(t) = (t + )(t + 2)(2t + 3)/6 c = 6 ν = (7532) P(t) = (t + ) 2 (t + 2)/2 c = 7 ν = (7432) P(t) = (t + )(t + 2)(t 2 + 3t + 6)/6 Seoul - 28 p. 82/86

83 Generating function for LR-polynomials Ex: Let F(z) = G(z)/( z) d+ = t= P(t) zt. c = ν = (953) d = G(z) = c = 2 ν = (943) d = 2 G(z) = c = 3 ν = (844) d = 3 G(z) = c = 4 ν = (853) d = 4 G(z) = c = 4 ν = (7442) c = 5 ν = (754) c = 6 ν = (7532) d = 3 G(z) = + z d = 4 G(z) = + z d = 4 G(z) = + 2z c = 7 ν = (7432) d = 5 G(z) = + 2z + z 2 Seoul - 28 p. 83/86

84 Further example Ex: n = 7, λ = (4332), µ = (4322), ν = (744432). LR coefficient c ν λµ = 3 LR polynomial c tν tλ.tµ = /8 where 8 = 5! 84 (t + )(t + 2)(t + 3)(t + 4)(t + 5) (5t + 2)(t 2 + 2t + 4) d = 8 and G(z) = + 4z + 2z 2 + 3z 3 Seoul - 28 p. 84/86

85 Contents Schur functions and LR-coefficients The hive model - vertex and edge labelling Horn inequalities Factorisation Problems of symmetry, conjugacy and complementarity Inequalities of the Stembridge and Gutschwager type Stretched LR-coefficients and LR-polynomials Open problems - positivity of coefficients, degrees and zeros of LR-polynomials Seoul - 28 p. 85/86

86 Polynomial behaviour Theorem For all λ,µ,ν such that c ν λµ > there exists a polynomial P ν λµ (t) in t with P ν λµ () = such that Pλµ ν (t) = ctν tλ,tµ Conjectures for all positive integers t. coefficients in Pλµ ν (t) are all rational and non-negative. coefficients in G(z) are all positive integers. Problems predict degree of polynomial explain origin of factors of form (t + )(t + 2) (t + m) prove (if true) and account for positivity of coefficients Seoul - 28 p. 86/86