The Fibonacci partition triangles R 1 R 2 R 3 R 4 R d 3 (t)d 4 (t)d 5 (t)d 6 (t)

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1 The Fibonacci partition triangles 0 0 P P P P P 0 f f f f f 0 f f f f f 0 f f f R R R R R f f f f f f f f f f f f f 0 0 d (t)d (t)d (t)d (t)d (t) d (t)d (t)d (t)d (t) d (t) d (t) d (t) d (t) t The even-index triangle The odd-index triangle

2 - Banff (0) Some Cartan data: some Kac-Moody Lie-algebras First: n-kronecker Second: n-regular tree Third: Non-symmetric lines (two are needed) () The Fibonacci partition triangles and their relationship - Construction, as addition function on a valued translation quiver mesh-additivity; equivalently: hook additivity - The Fibonacci-formulae - Relationship: Differences along south-west arrows (thus, for the even-index triangle, also along south-east arrows) Conversely: sum formulae Crossing differences left-right relationship for the odd-index triangle N-rules - Pylonmiality - Delannoy numbers () The reduction steps Lukas number L n = F n +F n+ Theorem Let (U, V, α, β, γ) be an indecomposable -Kronecker module The either dimu/dimv φ = φ + (with φ = ( + )/ the golden section), or else dimu = f n,dimv = f n+ (and therefore dimu +dimv = The modules which we call Fibonacci modules: Even index: The unique indecomposable -Kronecker module with dimension vector (f n+,f n ), for n 0, thus (f,f 0 ),(f,f ), and so on, or the dual ones (with dimension vectors (f n,f n+ ) Oddindex: Theindecomposable-Kroneckermoduleswithdimensionvector(f n+,f n ), and (f n,f n+ ), in both cases there is a -parameter family, indexed by P, and in both cases there are precisely isomorphism classes of tree modules Recall that an indecomposable representation of a quiver is called a tree module, provided is has (total) dimension d and can be described my matrices involving only d non-zero coefficients (these coefficients can then we chosen to be equal to ) The Fibonacci modules which are tree modules are obtained from indecomposable representations of the -regular tree with bipartite orientation (this is called covering theory, it is a non-commutative version of dealing with graded modules - modules graded by the elements of a free non-abelian group, here the free group in generators) Typical example: P with dimension vector (f,f 0 ) = (,), with a typical arm being of the form (we have = + ( + ) and = ( + + ), this we call a Fibonacci partition formula) Main question: How to define the subspaces for the preprojective -Kronecker representations which yield the covering?

3 For P = (M,M ) we take just M and the images of α,β,γ this yields a the -dim projective D -representation for the factor-space orientation Next: P Reconditioning: Separation(grading,Z-gradingorZ n -grading)wegetthen-kronecker algebra Reconditioning : Universal cover (grading by a free group) we get the path category of the n-regular tree Reconditioning : working modulo the symmetries (or some symmetries) which fix a base point (the group G x ) or a neighboring pair (the group G xy ) we deal with a valued tree (ray ev or line odd ) Covering theory and hammocks: how to recover the graded pieces of the given vector spaces? Relationship to finitely definable subspaces To be used for calculating Hall polynomials! () Categorification () The relevance of the Fibonacci numbers in representation theory (non-commuative geometry) Finite length modules: semisimple modules, modules of Loewy length at most Recall: If R is a commutative noetherian ring, then Ext (S,S ) = 0 for S,S simple and not isomorphic Question: is it true for general commutative rings? ) Non-commutative algebra for example: study of length categories difference commutative rings - non-comm rings: extensions of different simple modules localization, simplification n-kronecker algebra Relevance: describes the extensions of two simple modules Relevant even for the commutative theory: k[x,y,z]/(x,y,z) Study of (Z)-graded modules: Use of non-commutative phenomena! ) The structure of the module category for the n-kronecker algebra Real modules, real roots n = : natural numbers n = : Fibonacci numbers In analogy to the duality between affine schemes and commutative rings, we define the category of noncommutative affine schemes as the dual of the category of associative unital rings van Oystaeyen, Algebraic geometry for associative algebras A L Rosenberg, Noncommutative schemes, Compositio Math ()

4 Relevance of the Fibonacci numbers in mathematics Here: in the representation theory of fin dim algebras Main Theorem: Trichotomy: finite - tame - wild Tame: one-parameter families tubular families, indexed by the natural numbers Typical: Kronecker Wild: Minimal wild, Typical -Kronecker Relevant condition: finite dimensionality Otherwise also polynomial features, Example: k[x, Y] relationship is encoded by the so-called rep dimension Back to fin-dim: Existence of projective reps, then preprojective reps (more general: Take-off subcategories) Tree modules: use of bases relevance of the -regular tree Exponentielles Wachstum: Binets Formel besagt: f n /sqrt()((+sqrt())/) n < /,thusf n isthenearestintegertothevaluesoftheexponentialfunctionn /sqrt()((+ sqrt())/) n Minimal wild subquivers of the -regular tree: the usual minimal wild one-point extensions of D,Ẽm with m =,, (a quiver with,, 0 vertices) (better: Dn with n (a quiver with,,,0 vertices) This means: up to duality, there are seven cases How do we get the -Kronecker? Let ω be an extension vertex of the affine quiver, take X = P(ω)+H, this yields an indecomposable preprojective representation (thus a brick) for the affine quiver; now take the further one-point extension using M = P(ω), with extension vertex ω Then X and S(ω ) are othogonal bricks with dimext (S(ω ),X) = The growth number for some problems: a) The preprojective -Kronecker quiver: φ with φ = ( + )/,0, and φ, b) Central binomial coefficient: binom(n,n) should be = c) the pylon of the even-index triangle d) the pylon of the truncated even-index triangles (with only finitely many columns, say the last r) () Further consideration: Pisano-periods Theorem: F n is divisible by iff n is divisible by

5 Proof: Take the F n modulo, the Pisano period is, this sequence is of the form x x x x x x x x 0 x x x x x x x x 0 x x x x x x x x 0 x x x x x x x x 0 with x 0 Corollary: If divisible by, then by 0 Corrections to part p, line Corollary The function d i (t) with t i Remark: Even for t i, but not for t = i, if t Namely, by definition, we put d i (i ) = d i (i ) and??? On the other hand, for example, for i = and p (t) = d (t) for t large, then p (t) = (t +t ), we have d () = d 0 () =, but p () = 0 In general, we have for the polynomial p i with p i (t) = d i (t) for large t: p i+ (i) = d i (i) d i (i), as the mesh relations show Now if p i+ (i) = d i (i), then d i+ (i) = d i (i), which is never the case line : Replace: Thus g i by: Thus d i The formula for d is d (t) = (t +t ) = ( ) t + ( ) t ( ) t 0 It follows that the function d i is determined by the functions d 0,,d i as well as one special value, for example the pylon value d i (i) In this way, we see that the pylon values determine the whole triangle d i In section : In Corollary : The functions d i (t) with t i and the functions (t) with t i+ are polynomials, for all i As for the even-index triangle, we see again that the pylon numbers (for one of the two pylons!) determine completely all the other values of the triangle

6 Divisibility by The Fibonacci partition triangles II Proposition In the even-index triangle, the numbers on the pylon are not divisible by Proof: These numbers add up to f n+, but = f divides f m iff divides m, thus does not divide f n+ The Fibonacci partitionformula asserts that divides thedifference between f n+ and the pylon number, thus does not divide the pylon number Divisibility by is really related to the -fold symmetry of the -regular tree

7 Combining hook formulae For i, d i (i) = d i (i )+ 0 j i d j(i+j) This means: 0 Proof: We start with the hook formula for d i (i) d i (i) = d i (i )+ d j (i+j) 0 j i = d i (i )+ 0 j i d j (i+j) and use the hook formula for d i (i ), namely d i (i ) = d i (i )+ d j (i+j) 0 j i

8 0 Properties of the Fibonacci modules For the even-index Fibonacci modules and the odd-index ones R i with i, the arms end in n

9 The -regular tree with subspace orientation (one sink, no ray) We call a representation of the -regular tree with subspace orientation (one sink, no ray) a subspace system The even-index triangle yields all the entries of the dimension vectors of the preprojective representations Note that for this quiver, the indecomposable projectives are serial modules (of arbitrary length) Of particular interest are those in the τ -orbit of the indecomposable projectives of length : the corresponding total space is given by a Delannoy number (with the noncrossing hypothesis) Relevance () If we deal with a quiver Q with cover Q, and start with an indecomposable representation M of Q with push down π(m) How to recover the vector spaces of M starting from π(m) () If M is an indecomposable representation of a quiver, and x is a vertex of Q Is M x intrinsically an indecomposable S-space for some poset S (as one knows for representation-directed algebras, according to the hammock approach) For (), we need a generailzation of the hammock-approach to the representationinfinite case In our case (the Fibonacci modules), we can formulate a precise recipe Example: Take the number in the even-index triangle (it occurs just once), it corresponds to the subspace representation How do we see this in the triangle?? Here, we deal with M(,) Let P(t) an indecomposable projective subspace system of length t+, thus an indecomposable projective subspace system generated at a vertex z with D(x, z) = t Theorem Let M i,t be the vector space at the position (i,t) in the even-index triangle M (i,t) τ i P(t) Proof: Start with P(t) and apply reflection functors in order to get P i P(t) Continue applying reflection functors in order to get bipartite orientation, and not changing the orientation at (i, t)

10 Polynomials Here are the polynomials d,d,d : d (t) = t + t t t+ ( ) ( ) ( ) t t t = + t+, d (t) = 0 t + t t t + 0 t+ ( ) ( ) ( ) t t t = + +t+, d (t) = 0 t + t t t + 0 t + t ( ) ( ) ( ) ( ) ( ) t t t t t = + + +t 0

11 Which formulae are categorified - which not Cassini: F n Fn+ F n = ( )n

12 Pisano periods Claim: The Pisano period modulo m is at most m and it is precisely m if and only if m = t for some t See:

13 The structure of the Fibonacci-modules P t (x) The numbers in the even-index triangle are precisely those in the dimension vectors of the Coxeter shifts τ t P of the indecomposable projective modules P for the subspace orientation (Note that such a P is serial and corresponds to a path to the unique sink) The Delannoy modules (the highest number is a Delannoy number) are those of the form τ t P with P of length )

14 Covering theory and pp-subspaces We want to describe the covering decomposition of the Fibonacci modules in terms of pp-subspaces (ie pp-definable subgroups in the terminology of Prest, ) Examples: P : the center is the next circle is C = (αm +βm) (βm +γm) (γm +αm) α C, β C, γ C, the outer circle are certain images of the first circle: βα C,γα C, Consider P with walk the center is α C = ( α (βm +γm)+β (γm +αm) ) ( ) ( ) The first circle is (αc +βm) (αc +γm),, The second circle starts with: β ((αc +βm) (αc +γm)) The thrid circle starts with αβ ((αc +βm) (αc +γm)) Consider P with walk α

15 The even-index Fibonacci partition triangle valuation: (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) f 0 P f P f P f P f 0 P f P f P f P 0 f P f 0 P 0 0 f 0 f 0 f d 0 (t) d (t) d (t) d (t) d (t) d (t) d (t) f t+ t

16 The odd-index Fibonacci partition triangle valuation: (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) f t+ f t 0 R f R f R f R f R f f f 0 f f 0 f 0 f f d (t) d (t) d (t) d (t) d (t) d (t) d (t) d (t) d (t) t