Brauer Configuration Algebras and Matrix Problems to Categorify Integer Sequences
Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, Isaías David Marín Gaviria, Gabriel Bravo Rios
aa r X i v : . [ m a t h . R T ] F e b Brauer Configuration Algebras and Matrix Problems to CategorifyInteger Sequences
Agust´ın Moreno Ca˜nadas (corresponding author)Pedro Fernando Fern´andez EspinosaIsa´ıas David Mar´ın GaviriaGabriel Bravo Rios
Abstract
Bijections between invariants associated to indecomposable projective modules oversome suitable Brauer configuration algebras and invariants associated to solutions ofthe Kronecker problem and the four subspace problem are used to categorify integersequences in the sense of Ringel and Fahr. Dimensions of the Brauer configurationalgebras and their corresponding centers involved in the different processes are givenas well.
Keywords and phrases : Auslander-Reiten quiver, Brauer configuration algebra, cate-gorification, four subspace problem, indecomposable representation, integer sequence,Kronecker problem, A100705, A052591, OEIS.Mathematics Subject Classification 2010 : 16G20; 16G60; 16G30.
According to Ringel and Fahr [6] a categorification of a sequence of numbers means toconsider instead of these numbers suitable objects in a category (for instance, repre-sentation of quivers) so that the numbers in question occur as invariants of the objects,equality of numbers may be visualized by isomorphisms of objects functional relationsby functorial ties. The notion of this kind of categorification arose from the use of suit-able arrays of numbers to obtain integer partitions of dimensions of indecomposablepreprojective modules over the 3-Kronecker algebra (see Figure (1) where it is shownthe 3-Kronecker quiver and a piece of the oriented 3-regular tree or universal covering(
T, E, Ω t ) as described by Ringel and Fahr in [5]). Firstly they noted that the vectordimension of these kind of modules consists of even-index Fibonacci numbers (denoted f i and such that f i = f i − + f i − , for i ≥ f = 0, f = 1) then they used resultsfrom the universal covering theory developed by Gabriel and his students to identifysuch Fibonacci numbers with dimensions of representations of the corresponding uni-versal covering. In particular, preinjective and preprojective representations of the3-Kronecker quiver were used in [5] by Ringel and Fahr in order to derive a partitionformula for even-index Fibonacci numbers f n . A.M. Ca˜nadas et al ◦ ◦ z z d d o o • • (cid:127) (cid:127) ⑦⑦⑦ • (cid:31) (cid:31) ❅❅❅ • _ _ ❅❅❅ (cid:127) (cid:127) ⑦⑦⑦ / / • •• • _ _ ❅❅❅ ? ? ⑦⑦⑦ (cid:15) (cid:15) • (cid:127) (cid:127) ⑦⑦⑦ (cid:31) (cid:31) ❅❅❅ O O •• • • o o ? ? ⑦⑦⑦ (cid:31) (cid:31) ❅❅❅ • _ _ ❅❅❅ • ? ? ⑦⑦⑦ • (1)For the sake of clarity we give here a brief insight into the program of Ringel and Fahr.First of all note that the road to a categorification of the Fibonacci numbers hasseveral stops some of them dealing with diagonal (lower) arrays of numbers of theform D = ( d i,j ) with 0 ≤ j ≤ i ≤ n , (columns numbered from right to the left, seearray (5)) for some n ≥ d i,i = 1 , for all i ≥ ,d , = 2 ,d t +1 , = 0 , for all t ≥ ,d i +2 t,i − = 0 , if i ≥ , t ≥ , d i,j + d i,j − − d i − ,j − = d i +1 ,j − , i, j ≥ . (2)Besides, if i ≥ i − X t =0 d i + t,i − t + d i − , = d i − , . (3)Note that to each entry d i,i − j it is possible to assign a weight w i,i − j such that: w i,i − j = . ⌊ i − j ⌋ + a , if j is even , i = j, , if j is odd , i = j, , if i = j = 2 h for some h ≥ . Where ⌊ x ⌋ is the greatest integer number less than x , a ∈ { , − } , a = − i is even,it is 0 otherwise.The first stop consists of defining partitions of the even-index Fibonacci numbers inthe following form: f i +2 = i X j =0 ( w i,i − j )( d i,i − j ) , (4)to do that, Ringel and Fahr interpreted weights w i,i − j as distances in a 3-regular tree( T, E ) (with T a vertex set and E a set of edges) from a fixed point x ∈ T to anypoint y ∈ T . They define sets T r whose points have distance r to x , in such a case rauer Configuration Algebras... T = { x } , T are the neighbors of x and so on. A given vertex y is said to be evenor odd according to this parity [5].Any vertex y ∈ T yields a suitable reflection σ y on the set of functions T → Z with finite support, denoted Z [ T ], and some reflection products denoted Φ and Φ according to the parity of y are introduced in [5]. Then some maps a t : N → Z ∈ Z [ T ]are defined in such a way that if a is the characteristic function of T then a ( x ) = 0 unless x = x in which case a ( x ) = 1, and a t = (Φ Φ ) t a , for t ≥ a t [ r ] = a t ( x ), for r ∈ N and x ∈ T r , these maps a t give the values d i,j of thearray (2). The following table (called the even-index Fibonacci partition triangle) isan example of such array with n = 12. Rows are giving by the values of t , P t isa notation for a 3-Kronecker preprojective module with dimension vector [ f t +2 f t ](see [7]). (5)0123456789101112 tf f f f f f f f f f f f f f t +2 (2(6)+18)-5 P P P P P P P P P P a t [0] a t [1] a t [2] a t [3] a t [4] a t [5] · · · For example for t = 3 and t = 4, we compute f and f as follows: A.M. Ca˜nadas et al
21 = f = 0 + 3(3 . ) + 0 + 1(3 . ) ,
55 = f = 1 . . ) + 0 + 1(3 . ) . (6)Sequences a t [0] = d i, and a t [1] = d i +1 , are encoded respectively as A132262 andA110122 in the OEIS (On-Line Encyclopedia of Integer Sequences). Actually, sequence a t [0] had not been registered in the OEIS before the publication of Ringel and Fahr.In a second stop of the trip to a categorification of Fibonacci numbers, Ringel and Fahrgeneralized the results obtained in [5], and proved that the following exact sequences(7), (8) and filtration (9) are categorifications of identities (10). Where, for t ≥ P t ( x ) and R t ( x, y ) are indecomposable representations of the quiver Q = ( T, E, Ω xt )(Ω xt is a bipartite orientation such that x is a sink in case t is even and a source in case t is odd) for which s t ( x ) and r t ( x, y ) denote respectively their corresponding dimensionvectors, assuming that for even t the vertex x is a sink, and that for t odd the vertex x is a source. In this setting, the sequences x , x , . . . , x t and x − , x , . . . , x t , x t +1 denote suitable paths with x as a sink and z i being a neighbor of x i different from x i − and x i +1 . 0 → P t − ( y ) → P t ( x ) → R t ( x, y ) → → P t − ( y ′ ) → R t ( x, y ) → R t − ( y ′′ , x ) → . (7)0 → P ( z ) ⊕ · · · ⊕ P t ( z t ) → R t +1 ( x t , x t +1 ) → R ( x − , x ) . (8) P ( x ) ⊂ P ( x ) ⊂ · · · ⊂ P t ( x t ) with factors P i ( x i ) /P i − ( x i − ) = R i ( x i , x i − ) , ≤ i ≤ t. (9) f t +1 = f t − + f t ,f t +1 = 1 + t X i =1 f i and f t = t X i =1 f i − . (10)Note that the Auslander-Reiten sequences0 → P n − → P n → P n +1 → → R n − ,λ → E ( n, λ ) → R n +1 ,λ → E ( n, λ ) is an indecomposable module having dimension vector 3(dim R ( n, λ ))are categorifications of the identity f t − + f t +2 = 3 f t . (12)In a third stop of the road to a categorification of Fibonacci numbers Ringel and Fahr[7] named the array (2) a Fibonacci triangle and stated that its entries (nonzero entries) rauer Configuration Algebras... are categorified by the modules P n = P n ( x ) (called Fibonacci modules) provided thatsuch entries give the Jordan-H¨older multiplicities of these modules.Finally, we point out that Ringel in [14] exhibits combinatorial data which can bederived from a category Ind Λ, where Λ is a hereditary artin algebra of Dynkin type ∆and Ind Λ is a set of indecomposable Λ-modules. He comments that many enumerationproblems give rise to categorification of different integer sequences. For instance, thenumber of some tilting modules and the number of antichains in mod Λ categorify theCatalan numbers if Λ is an algebra of Dynkin type A n . Whereas, if Λ is of Dynkintype B n then such number of modules and antichains categorify the sequence (cid:0) nn (cid:1) .According to Ringel [14], the number of antichains in mod Λ as well as the number oftilting modules and the number of indecomposable Λ-modules are examples of Dynkinfunctions which do not depend on the orientation. Regarding, this particular kind offunctions he proposes to build an On-Line Encyclopedia of Dynkin Functions (OEDF)with the same purposes as the OEIS.In this work, in order to categorify integer sequences, we identify combinatorial infor-mation arising from the preprojective components of the 2-Kronecker algebra (or sim-ply the Kronecker algebra) and the tetrad (or four incomparable points) with combina-torial information arising from indecomposable projective modules over some Brauerconfiguration algebras introduced recently by Green and Schroll [9]. In particular, weuse these settings to define categorifications of the sequences encoded in the OEIS asA052558, A052591 and A100705. Such Brauer configuration algebras are defined byconfigurations of some multisets called polygons.We recall here that the Kronecker problem is equivalent to the problem of deter-mining the indecomposable representations over a field k of the following quiver Q (2-Kronecker quiver): Q = ◦ ◦ α v v β h h (13)whereas to determine the indecomposable representations of four incomparable points(a tetrad) is a very well known matrix problem named the four subspace problem(FSP). The solution of this problem is essentially equivalent to determine all of theindecomposable representations of the four subspace quiver F with the following form[8, 13, 17, 19]: ◦ (cid:15) (cid:15) F = ◦ / / ◦ ◦ o o ◦ O O (14) A.M. Ca˜nadas et al
We will see that some invariants associated to indecomposable projective modules oversome suitable Brauer configuration algebras allow categorifications of integer numbers.In fact, since polygons in Brauer configurations are multisets, we will often assumethat such polygons consists of words of the form w = x s x s . . . x s t − t − x s t t (15)where for each i , 1 ≤ i ≤ t , x i is an element of the polygon called vertex and s i is thenumber of times that the vertex x i occurs in the polygon. In particular, if vertices x i in a polygon V of a Brauer configuration are integer numbers then the correspondingword w will be interpreted as a partition of an integer number n V associated to thepolygon V where it is assumed that each vertex x i is a part of the partition and s i isthe number of times that the part x i occurs in the partition and n V = t P i =1 s i x i .In this paper, we prove the following results: Theorem A.
Let ( P, A, B, n ) , ( P ′ , A ′ , B ′ , n ) , H P and H P ′ be two matrices of type H n with corresponding sets of helices H P and H P ′ defined by systems of the form ( i P , j P , P A , P B ) and ( f P ′ , g P ′ , P ′ A ′ , P ′ B ′ ) , respectively. Then | H P | = h Pn = | H P ′ | = h P ′ n . Helices associated to suitable ( n +1) × n -matrices of type H n are introduced in section2.2. Corollary B.
If for n ≥ , P and P ′ are equivalent preprojective Kronecker moduleswith dimension vector of the form [ n + 1 n ] and corresponding sets of helices H P and H P ′ then | H P | = | H P ′ | . Theorem C.
If for n ≥ , P denotes a preprojective Kronecker module then thenumber of helices associated to P is h Pn = n ! ⌈ n ⌉ where ⌈ x ⌉ denotes the smallest integergreatest than x . For a suitable Brauer configuration algebra Λ K n , the following result categorifies inthe sense of Ringel and Fahr the number of helices associated to some preprojectiveKronecker modules. Corollary D.
For n ≥ fixed and ≤ t ≤ n , the number of summands in the heart ofthe indecomposable projective representation V t over the Brauer configuration algebra Λ K n equals the number of helices associated to the preprojective Kronecker module (2 t + 3 , t + 2) , ≤ t ≤ n . Proposition E. If W P is the set of matrix words associated to a matrix P of type H n then | W P | equals n P m =0 P ( n, n, m ) = ( n + 1) C n , where P ( n, n, m ) denotes the number ofpartitions of m into n parts, each ≤ n , P ( n, n,
0) = 1 , and C n denotes the n th Catalannumber. The next theorem regards the number of cycles associated to preprojective represen-tations of the tetrad and its relationship with the number of summands in the heart ofan indecomposable module over a suitable Brauer configuration algebra. The notionof cycle associated to suitable (2 n + 2) × (4 n + 3)-matrices of type C n is introduced insection 2.3. rauer Configuration Algebras... Theorem F.
For j ≥ fixed and ≤ i ≤ j , the number of summands in the heart ofthe indecomposable projective module T P i over the Brauer configuration algebra Λ E j equals the number of cycles associated to a preprojective representation of the tetradof type IV and order i + 1 . The following result gives a way to categorify any counting function u n , i.e., sequenceswhose elements count a given class of objects. Catalan numbers, Fibonacci numbers,Delannoy numbers and Dedekind numbers are some of the most well known countingfunctions. D n is a suitable Brauer configuration. Theorem G.
For ≤ i ≤ n and n > fixed, the number of summands in the heartof the indecomposable projective module P i over the Brauer configuration algebra Λ D n is u i . Dimension of the Brauer configuration algebras involved in the different results andtheir corresponding centers are given in Corollaries 13, 14, 23, 24 and Theorems 20,21, 27 and 28.This paper is distributed as follows: In section 2, we recall main definitions and no-tation used throughout the document, in particular, in this section we define Brauerconfiguration algebras, the Kronecker problem and the four subspace problem.In section 3, in order to categorify numbers in sequences A052558 and A052591, weprove that numbers in these sequences give the number of some helices associated topreprojective Kronecker modules. Besides, it is defined a sequence Λ K n of Brauerconfiguration algebras whose indecomposable projective modules are in bijective cor-respondence with preprojective Kronecker modules via the number of summands inthe heart of such indecomposable modules. Formulas for the dimension of this type ofalgebras and corresponding centers are given as well.In section 4, numbers in the sequence A100705 are categorified by determining thenumber of cycles associated to preprojective representations of type IV of the tetradand it is introduced a sequence Λ E n of Brauer configuration algebras such that thenumber of summands in the heart of their indecomposable projective modules coincideswith the number of cycles associated to the mentioned preprojective representations.The dimension of these algebras and corresponding quotients are also obtained in thissection.In section 5, we describe how it is possible to use integer sequences in order to buildBrauer configuration algebras, the process is applied to any counting function and inparticular to the sequence A100705. Examples of helices are given in section 6. In this section, we recall main definitions and notation to be used throughout thepaper [1, 4, 9, 19].
A.M. Ca˜nadas et al
Brauer configuration algebras were introduced by Green and Schroll in [9] as a gener-alization of Brauer graph algebras which are biserial algebras of tame representationtype and whose representation theory is encoded by some combinatorial data based ongraphs. According to them, underlying every Brauer graph algebra is a finite graphwith a cyclic orientation of the edges at every vertex and a multiplicity function [15].The construction of a Brauer graph algebra is a special case of the construction of aBrauer configuration algebra in the sense that every Brauer graph is a Brauer con-figuration with the restriction that every polygon is a set with two vertices. In thesequel, we give precise definitions of a Brauer configuration and a Brauer configurationalgebra.A
Brauer configuration
Γ is a quadruple of the form Γ = (Γ , Γ , µ, O ) where:( B
1) Γ is a finite set whose elements are called vertices ,( B
2) Γ is a finite collection of multisets called polygons . In this case, if V ∈ Γ thenthe elements of V are vertices possibly with repetitions, occ( α, V ) denotes thefrequency of the vertex α in the polygon V and the valency of α denoted val ( α )is defined in such a way that: val ( α ) = X V ∈ Γ occ( α, V ) , (16)( B µ is an integer valued function such that µ : Γ → N where N denotes the set ofpositive integers, it is called the multiplicity function ,( B O denotes an orientation defined on Γ which is a choice, for each vertex α ∈ Γ ,of a cyclic ordering of the polygons in which α occurs as a vertex, includingrepetitions, we denote S α such collection of polygons. More specifically, if S α = { V ( α )1 , V ( α )2 , . . . , V ( α t ) t } is the collection of polygons where the vertex α occurswith α i = occ( α, V i ) and V ( α i ) i meaning that S α has α i copies of V i then anorientation O is obtained by endowing a linear order < to S α and adding arelation V t < V , if V = min S α and V t = max S α . According to this order the α i copies of V i can be ordered as V ,i < V ,i < · · · < V ( α i − ,i < V α i ,i and S α can be ordered in the form V ( α )1 < V ( α )2 < · · · < V ( α ( t − )( t − < V α t t ,( B
5) Every vertex in Γ is a vertex in at least one polygon in Γ ,( B
6) Every polygon has at least two vertices,( B
7) Every polygon in Γ has at least one vertex α such that µ ( α ) val ( α ) > S α , < ) is called the successor sequence at the vertex α .A vertex α ∈ Γ is said to be truncated if val ( α ) µ ( α ) = 1, that is, α is truncated if itoccurs exactly once in exactly one V ∈ Γ and µ ( α ) = 1. A vertex is nontruncated ifit is not truncated. The Quiver of a Brauer Configuration Algebra rauer Configuration Algebras... The quiver Q Γ = (( Q Γ ) , ( Q Γ ) ) of a Brauer configuration algebra is defined in sucha way that the vertex set ( Q Γ ) = { v , v , . . . , v m } of Q Γ is in correspondence withthe set of polygons { V , V , . . . , V m } in Γ , noting that there is one vertex in ( Q Γ ) forevery polygon in Γ .Arrows in Q Γ are defined by the successor sequences. That is, there is an arrow v i s i −→ v i +1 ∈ ( Q Γ ) provided that V i < V i +1 in ( S α , < ) ∪ { V t < V } for some nontruncatedvertex α ∈ Γ . In other words, for each nontruncated vertex α ∈ Γ and each successor V ′ of V at α , there is an arrow from v to v ′ in Q Γ where v and v ′ are the vertices in Q Γ associated to the polygons V and V ′ in Γ , respectively. The Ideal of Relations and Definition of a Brauer Configuration Algebra
Fix a polygon V ∈ Γ and suppose that occ( α, V ) = t ≥ t in-dices i , . . . , i t such that V = V i j . Then the special α -cycles at v are the cycles C i , C i , . . . , C i t where v is the vertex in the quiver of Q Γ associated to the polygon V . If α occurs only once in V and µ ( α ) = 1 then there is only one special α -cycle at v .Let k be a field and Γ a Brauer configuration. The Brauer configuration algebraassociated to
Γ is defined to be the bounded path algebra Λ Γ = kQ Γ /I Γ , where Q Γ isthe quiver associated to Γ and I Γ is the ideal in kQ Γ generated by the following set ofrelations ρ Γ of type I, II and III.1. Relations of type I . For each polygon V = { α , . . . , α m } ∈ Γ and each pairof nontruncated vertices α i and α j in V , the set of relations ρ Γ contains allrelations of the form C µ ( α i ) − C ′ µ ( α j ) where C is a special α i -cycle and C ′ is aspecial α j -cycle.2. Relations of type II . Relations of type II are all paths of the form C µ ( α ) a where C is a special α -cycle and a is the first arrow in C .3. Relations of type III . These relations are quadratic monomial relations of theform ab in kQ Γ where ab is not a subpath of any special cycle unless a = b and a is a loop associated to a vertex of valency 1 and µ ( α ) > I and ρ insteadof Λ Γ , I Γ and ρ Γ for a Brauer configuration algebra, the ideal and set of relations,respectively defined by a given Brauer configuration Γ.As a toy example consider a configuration Γ = (Γ , Γ , µ, O ) such that:1. Γ = { , , , } ,2. Γ = { U = { , , , , , } , V = { , , , , }} ,3. At vertex 1, it holds that; S = { U (2) V (1) } , U < U < V , val (1) = 3,4. At vertex 2, it holds that; S = { U (1) V (1) } , U < V , val (2) = 2,5. At vertex 3, it holds that; S = { U (2) V (1) } , U < U < V , val (3) = 3,6. At vertex 4, it holds that; S = { U (1) V (2) } , U < V < V , val (4) = 3,7. µ ( α ) = 1 for any vertex α . A.M. Ca˜nadas et al
The ideal I of the corresponding Brauer configuration algebra Λ Γ is generated by thefollowing relations (see Figure (18)), for which it is assumed the following notation forthe special cycles: P u, = c c c ,P u, = c c c ,P u, = c c ,P u, = c c c ,P u, = c c c ,P u, = c c c ,P v, = c c c ,P v, = c c ,P v, = c c c ,P v, = c c c ,P v, = c c c , (17)1. c hi c sr , if h = s , for all possible values of i and r ,2. ( c ) ; ( c ) ; ( c ) , c c , c c , c c ,3. P u,ij − P u,tl , for all possible values of i, j, t and l ,4. P v,ij − P v,tl , for all possible values of i, j, t and l ,5. P u,ji a ( P v,ji a ′ ) , with a ( a ′ ) being the first arrow of P u,ji ( P v,ji ) for all i, j .The following diagrams (18-24) show the quiver Q Γ associated to this configuration,the indecomposable projective modules P U and P V , corresponding heart and radicalsquare of these modules. Q Γ = /.-,()*+ U c M M c (cid:17) (cid:17) c , , c c (cid:25) (cid:25) c (cid:22) (cid:22) /.-,()*+ V c y y c l l c c c c Y Y c V V (18) rauer Configuration Algebras... P ( U ) : U c w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ c (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ c (cid:15) (cid:15) c (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅ c ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ c * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ V c (cid:15) (cid:15) U c (cid:15) (cid:15) U c (cid:15) (cid:15) V c (cid:15) (cid:15) V c (cid:15) (cid:15) V c (cid:15) (cid:15) U c ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ V c (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅ V c (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ U c w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ V c t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ U (19)Heart( P ( U )) : V c (cid:15) (cid:15) U c (cid:15) (cid:15) U c (cid:15) (cid:15) V c (cid:15) (cid:15) V c (cid:15) (cid:15) VU V V U V (20)rad P ( U ) : U c ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ V c (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅ V c (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ U c w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ V c t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ U (21) A.M. Ca˜nadas et al P ( V ) : V c w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ c (cid:7) (cid:7) ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎ c (cid:15) (cid:15) c (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅ c ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ U c (cid:15) (cid:15) U c (cid:15) (cid:15) V c (cid:15) (cid:15) U c (cid:15) (cid:15) U c (cid:23) (cid:23) ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ U c ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ U c (cid:15) (cid:15) U c (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ V c w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ V (22)Heart P ( V ) : U c (cid:15) (cid:15) U c (cid:15) (cid:15) V c (cid:15) (cid:15) U c (cid:15) (cid:15) UU U U V (23)rad P ( V ) : U c ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ U c (cid:15) (cid:15) U c (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ V c w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ V (24)The following results give some description of the structure of Brauer configurationalgebras [9, 16]. Theorem 1 ([9], Theorem B, Proposition 2.7, Theorem 3.10, Corollary 3.12) . Let Λ be a Brauer configuration algebra with Brauer configuration Γ.1.
There is a bijective correspondence between the set of indecomposable projective Λ -modules and the polygons in Γ.2. If P is an indecomposable projective Λ -module corresponding to a polygon V in Γ . Then rad P is a sum of r indecomposable uniserial modules, where r is thenumber of (nontruncated) vertices of V and where the intersection of any twoof the uniserial modules is a simple Λ -module .3. A Brauer configuration algebra is a multiserial algebra . rauer Configuration Algebras... The number of summands in the heart ht ( P ) = rad P/ soc P of an indecompos-able projective Λ -module P such that rad P = 0 equals the number of nontrun-cated vertices of the polygons in Γ corresponding to P counting repetitions .5. If Λ ′ is a Brauer configuration algebra obtained from Λ by removing a truncatedvertex of a polygon in Γ with d ≥ vertices then Λ is isomorphic to Λ ′ . Proposition 2 ([9], Proposition 3.3) . Let Λ be the Brauer configuration algebra as-sociated to the Brauer configuration Γ . For each V ∈ Γ choose a nontruncated vertex α and exactly one special α -cycle C V at V , A = { p | p is a proper prefix of some C µ ( α ) where C is a special α − cycle } ,B = { C µ ( α ) V | V ∈ Γ } . Then A ∪ B is a k -basis of Λ. Proposition 3 ([9], Proposition 3.13) . Let Λ be a Brauer configuration algebra associ-ated to the Brauer configuration Γ and let C = { C , . . . , C t } be a full set of equivalenceclass representatives of special cycles. Assume that for i = 1 , . . . , t , C i is a special α i -cycle where α i is a nontruncated vertex in Γ. Then dim k Λ = 2 | Q | + P C i ∈C | C i | ( n i | C i | − where | Q | denotes the number of vertices of Q , | C i | denotes the number of arrows inthe α i -cycle C i and n i = µ ( α i ).The following result regards the center of a Brauer configuration algebra. Theorem 4 ([16], Theorem 4.9) . Let Γ be a reduced and connected Brauer configu-ration and let Q be its induced quiver and let Λ be the induced Brauer configurationalgebra such that rad Λ = 0 then the dimension of the center of Λ denoted dim k Z (Λ) is given by the formula :dim k Z (Λ) = 1 + X α ∈ Γ µ ( α ) + | Γ | − | Γ | + Loops Q ) − | C Γ | . where | C Γ | = { α ∈ Γ | val ( α ) = 1 , and µ ( α ) > } .As an example the following are the numerical data associated to the algebra Λ Γ = kQ Γ /I with Q Γ as shown in Figure (18) and special cycles given in (17), ( | r ( Q Γ ) | isthe number of indecomposable projective modules, r U and r V denote the number ofsummands in the heart of the indecomposable projective modules P ( U ) and P ( V )).Note that, | C i | = val ( i ): | r ( Q Γ ) | = 2 ,r U = 6 , r V = 5 , | C | = 3 , | C | = 2 , | C | = 3 , | C | = 3 , X α ∈ Γ X X ∈ Γ occ( α, X ) = 11 , the number of special cycles , dim k Λ Γ = 4 + 3(3 −
1) + 3(3 −
1) + 2(2 −
1) + 3(3 −
1) = 24 , dim k Z (Λ Γ ) = 1 + 4 + (2 −
4) + 3 = 6 . (25) A.M. Ca˜nadas et al
The classification of indecomposable Kronecker modules was solved by Weierstrass in1867 for some particular cases and by Kronecker in 1890 for the complex number fieldcase. This problem is equivalent to the problem of finding canonical Jordan form ofpairs of matrices (
A, B ) (with the same size) with respect to the following elementarytransformations over a field k (for the sake of brevity, it is assumed that k is analgebraically closed field):(i) All elementary transformations on rows of the block matrix ( A, B ).(ii) All elementary transformations made simultaneously on columns of A and B having the same index number.If the matrix blocks P = ( A, B ) and P ′ = ( A ′ , B ′ ) can be transformed one into theother by means of elementary transformations, then they are said to be equivalent orisomorphic as Kronecker modules. The following is the matrix form (up to isomor-phism) of the non-regular Kronecker modules [17, 20]:II = III ∗ : → I n ← I n III = II ∗ : I ↑ n I ↓ n In this case, → I n ( ← I n , respectively) denotes an n × ( n + 1) matrix obtained from theidentity I n by adding a column of zeroes in fact the last column (the first column,respectively) in these matrices consists only of zeroes. In the same way, I ↑ n ( I ↓ n )denotes an n + 1 × n matrix obtained from an n × n identity matrix by adding at thetop (at the bottom) a row of zeroes.We recall that the solution of the Kronecker matrix problem allows to classify theindecomposable representations of the path algebra kQ with Q a quiver of the form(13).Figure (26) shows the preprojective component of the Auslander-Reiten quiver of the2-Kronecker quiver which has as vertices isomorphism classes of indecomposable repre-sentations of type III ([ i +1 i ] is a notation for the dimension vector of a preprojectiverepresentation (equivalently, preprojective module), whereas [ m m + 1] is the dimen-sion vector of a preinjective module). The preinjective component has isomorphismclasses of indecomposable representations of type III ∗ as vertices.[1 0] [3 2] [5 4][2 1] [4 3] [6 5] (cid:0)(cid:0)(cid:0)✒ (cid:0)(cid:0)(cid:0)✒ (cid:0)(cid:0)(cid:0)✒(cid:0)(cid:0)(cid:0)✒ (cid:0)(cid:0)(cid:0)✒ (cid:0)(cid:0)(cid:0)✒❅❅❅❘ ❅❅❅❘❅❅❅❘ ❅❅❅❘ ................ ................ ................................ ................ ................ (26)Henceforth, we let ( n + 1 , n ) (( n, n + 1)) denote a representative of an isomorphismclass of preprojective (preinjective) Kronecker modules obtained from a representation rauer Configuration Algebras... of type III (II) via elementary transformations. Actually, for the sake of simplicity, wewill assume that such representatives have the form III (II).For n ≥
1, let P be an ( n + 1) × n , k -matrix then P can be partitioned into two( n + 1) × n matrix blocks A and B . In such a case we write P = ( P, A, B, n ), where A = ( a i,j ) = [ C Ai , . . . , C Ai n ], B = ( b i,j ) = [ C Bj , . . . , C Bj n ], with C Ai r ( C Bj s ) columns of P ,if I A ( I B ) is the set of indices I A = { i r | ≤ r ≤ n } ( I B = { j s | ≤ s ≤ n } ) then I A ∩ I B = ∅ , and | I A | = | I B | = n . In this case, each column of the matrix P belongseither to the matrix A or to the matrix B and a word W P = l m . . . l m n . . . l m n , l m h ∈ { A, B } , 1 ≤ h ≤ n is used to denote matrix P by specifying the way thatcolumns of P have been assigned to the matrices A and B .A row r P of P has the form ( r A , r B ) with r A ( r B ) being a row of the matrix block A ( B ). We let R A ( R B ) denote the set of rows of the matrix block A ( B ), whereas H n denotes the set of all matrices P with the aforementioned properties.An helix associated to a matrix P of type H n is a connected directed graph h whoseconstruction goes as follows:( h ) ( Vertices ) Vertices of h are entries of blocks A and B . We let h denote the setof vertices of h .( h ) Fix two different rows i P = ( i A , i B ) and j P = ( j A , j B ) of P .( h ) Choose sets P A and P B of pivoting entries also called pivoting vertices , P A ⊂ A , P B ⊂ B such that | P A | = | P B | = n . Entries in A \ P A and B \ P B are said to be exterior entries or exterior vertices . In this case, if x ∈ P A ( x ∈ P B ) then x / ∈ i A ( x / ∈ j B ). P A and P B are sets of the form: P A = { a i ,j , a i ,j , . . . , a i s ,j s } , j x = j y if and only if i x = i y ,P B = { b t ,h , b t ,h , . . . , b t s ,h s } , h x = h y if and only if t x = t y . (27)Where, a i r ,j r ∈ R A \ i A , b t m ,h m ∈ R B \ j B , 1 ≤ r, m ≤ s . It is chosen just onlyone entry a i r ,j r ( b t m ,h m ) for each row in R A \ i A ( R B \ j B ) and for each column C A ( C B ) of A ( B ).( h ) ( Arrows ) arrows in h are defined in the following fashion;( a ) Arrows in h are either horizontal or vertical. We let h denote the set ofarrows of h .( b ) Horizontal arrows connect a vertex of the matrix block A ( B ) with a vertexof the matrix block B ( A ). Vertical arrows only connect vertices in the samematrix block. Starting and ending vertices of horizontal (vertical) arrows areentries of the same row (column) of P .( c ) The starting vertex of a horizontal (vertical) arrow is an exterior (pivot-ing) vertex. The ending point of a horizontal (vertical) arrow is a pivoting(exterior) vertex.( d ) A pivoting vertex occurs as ending (starting) vertex just once. Thus, h doesnot cross itself. A.M. Ca˜nadas et al ( e ) The first and last arrow of h are horizontal and its starting vertex belongsto i A .( f ) Each vertical arrow is preceded by a unique horizontal arrow, and unless thefirst arrow, any horizontal arrow is preceded by a vertical arrow.( g ) All the rows of P are visited by h , and no row or column of P is visited byarrows of h more than once.( h ) There are not horizontal arrows connecting exterior vertices of j A with ver-tices of j B . Remark 5.
We let ( i P , j P , P A , P B ) denote the set of all helices which can be built byfixing these data associated to a matrix P of type H n , h Pn = | ( i P , j P , P A , P B ) | denotesthe corresponding cardinality. See figures ((54), (56), (58), (60)) in section 6 where itis presented a set (4 P , P , P A , P B ) defined by the word BAABAB .Matrix presentations of preprojective Kronecker modules p of type III are of type H n , n ≥ W p = AA . . . ABB . . . B ). In [4], the first author et al studied setsof helices (1 p , ( n + 1) p , p A = { a i +1 ,i | ≤ i ≤ n } , p B = { b i,i | ≤ i ≤ n } ) associatedto this kind of matrices. Proposition 6. If W P is the set of matrix words associated to a matrix P of type H n then | W P | equals n P m =0 P ( n, n, m ) = ( n + 1) C n , where P ( n, n, m ) denotes the number ofpartitions of m into n parts, each ≤ n , P ( n, n,
0) = 1 , and C n denotes the n th Catalannumber. Proof.
Each matrix word W P of the form W P = l m . . . l m n . . . l m n , l m h ∈ { A, B } ,1 ≤ h ≤ n , gives rise to an integer partition λ = ( λ , λ , . . . , λ t ), λ i , t ≤ n of anonnegative integer number m ≤ n by defining λ as the number of A ’s after the firstoccurrence of the letter B , λ is the number of A ′ s after the second occurrence of theletter B and so on. Since there are n letters A ′ s and n letters B ′ s in W P then thenumber of words associated to P is (cid:0) nn (cid:1) . The result holds. (cid:3) In Theorem 10 we prove that the number of helices h pn associated to a preprojectiveKronecker module, p = ( n + 1 , n ) is h pn = n ! ⌈ n ⌉ .If we associate to a set of n equidistant points on a circle the rows of a representation p = ( n + 1 , n ) then the number of helices containing the fixed arrow a , → b , equalsthe number a ( n ) of ways of connecting n + 1 equally spaced points on a circle with apath of n line segments ignoring reflections. In this case, vertical edges in a helix arein bijective correspondence with the edges of the path in the circle (Figure (29) showsexamples of helices and this kind of paths). Thus a ( n ) = h pn n , n ≥ . (28) rauer Configuration Algebras... ,
1) = 0 ✲ ❄✛ ✣✢✤✜✣✢✤✜ • ❄ • (cid:0)(cid:0)(cid:0)✒ • ✲ • • (3 ,
2) = 010 001 010 100 ✲❄✛✻ ✲
Sequence a ( n ) is recorded as A052558 in the OEIS. The four subspace problem is another example of a matrix problem, in this case if k is an arbitrary field then a quadruple of finite-dimensional k vector spaces is a systemof the form U = ( U , U , U , U , U )where U is a finite- dimensional k vector space and U , . . . , U is an ordered collectionof four subspaces of U . Two quadruples are said to be isomorphic if there exists a k -space isomorphism ϕ : U → V such that ϕ ( U i ) = V i for all i . And a quadruple U is decomposable ( U = U ′ ⊕ U ′′ ) if some non-trivial direct sum decomposition U = U ′ ⊕ U ′′ satisfies the identity U i = ( U i ∩ U ′ ) ⊕ ( U i ∩ U ′′ ) for each i ∈ { , . . . , } .The four subspace problem consists of classifying all indecomposable quadruples upto isomorphism, it is equivalent to determine indecomposable representations of fourincomparable points or a tetrad. Actually, it is essentially equivalent to determine theindecomposable representations of the four subspace quiver (see Figure (14)).Given two matrix representations M = ( M x i , | ≤ i ≤
4) and M ′ = ( M ′ x i , | ≤ i ≤ M and M ′ are said to be equivalent or isomorphic, if one can beturned into the other by means of the following admissible transformations :1. k -elementary transformations of rows of the whole matrix.2. k -elementary transformations of columns of matrices M x i .FSP was solved by Gelfand and Ponomarev in 1970 for k algebraically closed andby Nazarova (1967-1973) for the arbitrary case. An advance to this problem wasgiven by Brenner who described the indecomposable quadruples with non-zero defect ∂ ( U ) = P i =1 dim U i − U (called non-regular) in particular she extended the resultsof Gelfand and Ponomarev to the case of a skew field k . Afterwards, in 2004 Zavadskijand Medina gave an elementary solution of this problem [2, 3, 8, 13, 19].For n ≥
1, we consider (2 n + 2) × (4 n + 3)-matrices of type C n , whose rows andcolumns are partitioned by four adjacent matrix blocks (from the left to the right), U , U , U and U denoted ( A , A ′ ); ( A , A ′ ); ( A , A ′ ) , ( A , A ′ ), respectively. A i and A ′ i are matrices vertically adjacent (blocks of type A i as well as those of type A ′ i A.M. Ca˜nadas et al are horizontally adjacent) of the same size, three of the four blocks U i consists of( n +1) × ( n +1)-matrices, and the remaining block consists of two ( n +1) × n -matrices.If U is a matrix of type C n then i th row ( j th column) i RU ( j CU ) of U is given by theunion i RU = ∪ m =1 i RA m , if i RU ⊂ ∪ m =1 A m , ∪ m =1 i RA ′ m , if i RU ⊂ ∪ m =1 A ′ m , where i RA m and i RA ′ m are corresponding rows in the matrices A m and A ′ m , m = 1 , . . . , j th column j CU of U is given by the union j CU = j CA s ∪ j CA ′ s if j CU ⊂ A s ∪ A ′ s , The following is a typical shape of a matrix U of type C n : U = A A A A A ′ A ′ A ′ A ′ Matrices of type C n have associated cycles which are connected oriented graph whoseconstruction goes as follows:( cl ) ( Vertices ) By definition any matrix U of type C n is defined by a word w whoseletters are the symbols U , U , U and U , i.e., w = U σ (1) U σ (2) U σ (3) U σ (4) where σ is a permutation of 4 elements. For the sake of clarity, later on, we assume thatthe matrix blocks of a matrix U of type C n are organized according to words ofthe form w = U σ (1) U σ (2) U σ (3) U σ (4) with σ ( i ) = i , 1 ≤ i ≤
4. Blocks U and U are said to be external blocks , whereas blocks U and U are internal blocks .( cl ) Fix three rows i RA i , j RA i and j ′ RA ′ i . In this case, U i is a (2 n + 2) × ( n + 1)-matrixblock, whereas U i is a (2 n + 2) × n -matrix block. Thus, according to our choice, i ∈ { , , } and i = 4.( cl ) Entries of matrix U of type C n are either pivoting or exterior vertices (or entries).They are defined as in the case for matrices of type H n , taking into account, thatin this case there is not a row i A to apply the restriction for the possible choicesof pivoting entries, and instead of j B , we consider rows j RA and j ′ RA ′ , which do notcontain pivoting entries. If P A rm denotes the set of pivoting entries of the matrix A rm , r = 0 , m = 1 , . . . , A i = A i , A i = A ′ i . Then | P A ri | = | P A rj | = n + 1,if i, j ∈ { , , } and | P A | = | P A ′ | = n .( cl ) Without loss of generality, we assume that the starting and ending vertex are thesame and belong to i RA ⊂ U , it is a pivoting vertex. Furthermore, j RA i ⊂ A and j ′ RA ′ i ⊂ A ′ .( cl ) According to the assumption introduced in ( cl ), the sequence of vertices con-nected by a cycle C is of the form, C = { a i ,j , a ′ i ′ ,j , b ′ i ′ ,j ′ , b i ,j ′ , c i ,j , c ′ i ′ ,j , d ′ i ′ ,j ′ } ∪ { d i ,j ′ , c i ,j , c ′ i ′ ,j , b ′ i ′ ,j ′ }∪ { b i ,j ′ , a i ,j } . Where entries ( a, a ′ ), ( b, b ′ ), ( c, c ′ ) and ( d, d ′ ) correspond respectively to theblocks ( A , A ′ ), ( A , A ′ ), ( A , A ′ ) and ( A , A ′ ). rauer Configuration Algebras... ( cl ) Any cycle contains the fixed vertices a i ,j , a ′ i ′ ,j , b ′ i ′ ,j ′ , b i ,j and c i ,j , whichis a pivoting entry. Entries of the form d ′ j ′ RAi ,j / ∈ C . No horizontal arrow hasan entry d j RAi ,j ∈ A as its ending vertex.( cl ) Vertices a i ,j ∈ A , b ′ i ′ ,j ′ ∈ A ′ , c i ,j ∈ A , d ′ i ′ ,j ′ ∈ A ′ , c i ,j ∈ A and b ′ i ′ ,j ′ ∈ A ′ are pivoting entries.( cl ) ( Arrows ) arrows in C connect alternatively pivoting entries with exterior entries.They are defined in the following fashion;( a ) Arrows in C are either horizontal ( → , ← ) or vertical ( ↑ , ↓ ). We let C denotethe set of arrows of C . We write, X → Y , X ← Y , X ↑ Y and X ↓ Y , thedifferent ways of connecting matrices of the different blocks U i . Accordingto the sequence of vertices, the first vertical arrow of the cycle C is of theform A ↓ A ′ .( b ) Since we are assuming cycles associated to a word of the form w = U U U U then horizontal arrows connect adjacent matrices A i ( A ′ i ) with A i +1 ( A ′ i +1 )(conversely, A i +1 ( A ′ i +1 ) with A i ( A ′ i )). The first horizontal arrow in ourcase is of the form A ′ → A ′ . Vertical arrows connect matrices in the form A i ↑ A ′ i or A ′ i ↓ A i . External blocks are connected by a unique vertical arrow(either A ↓ A ′ iff A ↑ A ′ or A ′ ↑ A iff A ↓ A ′ ). Two arrows, U → ← U , U → ← U connect internal matrix blocks ( A → A iff A ′ ← A ′ , A → A iff A ′ ← A ′ ). Two vertical arrows connect internal matrix blocks A ↓↓ A ′ or A ↑↑ A ′ , same conditions satisfy matrices A and A ′ .( c ) Each horizontal arrow is preceded by a unique vertical arrow, and unless thefirst arrow, any vertical arrow is preceded by an horizontal arrow.( d ) No row or column of a matrix A rm in a matrix block U m , r = 0 , m =1 , . . . , C more than once.As for matrices of type H n , any matrix U of type C n can be described in the form( U, U i , n | ≤ i ≤ i RA , j RA , j ′ RA ′ , P A ri | ≤ i ≤ , r ∈ { , } ) denotethe system of all cycles which can be constructed by using these data associated toa matrix U of type C n and C U = | ( i RA , j RA , j ′ RA ′ , P A ri | ≤ i ≤ , r ∈ { , } ) | arenotations for its corresponding cardinal.In this work, we compute the number of cycles associated to preprojective representa-tions of type IV of the tetrad and establish a connection between these preprojectiverepresentations with indecomposable projective modules over some Brauer configura-tion algebras via such cycles. The following is the canonical matrix presentation ofthe mentioned preprojective representations, where I n is an n × n identity matrix. n is said to be the order of the representation.I n +1 I n +1 I ↓ n n +1 n +1 I ↑ n (30)The following is an example of a cycle of type A.M. Ca˜nadas et al (1 RA , RA , RA ′ , P A ri = ∪ ≤ i ≤ r =0 , d ri ∪ d ∪ d ′ | r ∈ { , } )associated to a preprojective representation of order n = 3 of the tetrad, black arrowsconnect fixed vertices of the corresponding cycles, whereas d ri ( d ) denote the set ofdiagonal (subdiagonal) entries used to define the corresponding pivoting vertices. ❄ ✲✻ ✲ ❄ ✲✻✛❄✛✻✛ Binomial trees appear in many fields of the mathematics, they are binary trees withthe shape [10]: • T T · · · T n − ................................................................................................................................................ .................................................................................................................................... ............ ....................................................................................................................................................................................................................................................................................................................................................... ............ n − T has the following form: •• • • •• • • • • •• • •• ............................................................................................................. ................................................................................................. ............ ............................................................................................................................................................................. ............ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ............ .................................................................................................... ............................................................................................................. ................................................................................................. ............ .................................................................................................................................................. ............................................................................................................. ................................................................................................. ............ .................................................................................................... ............................................................................................................. ................................................................................................. ............ .................................................................................................... rauer Configuration Algebras... Note that, at each level T gives integer partitions of numbers 1, 2 and 3 (withouttaking into account 0 as a part). Often, these types of trees are said to be partitiontrees which can be used to store partitions of a given positive integer n or of allpositive integers ≤ n [11]. In [12], Luschny describes partition trees for differentinteger numbers and uses them to define some orders on the set of integer partitions. Results in this section can be interpreted as categorifications (see Remark 11) of thesequences ( n − ⌈ n ⌉ and n ! ⌈ n ⌉ (A052558 and A052591 in the OEIS, respectively) viaKronecker modules and Brauer configuration algebras. Theorems 7 and 10 and Corol-laries 8 and 9 prove that the number of helices associated to preprojective Kroneckermodules is invariant with respect to admissible transformations.The following results regard the number of helices associated to matrices of type H n and in particular to preprojective Kronecker modules (see Figure (26)), k is analgebraically closed field. Theorem 7.
Let ( P, A, B, n ) , ( P ′ , A ′ , B ′ , n ) , H P and H P ′ be two matrices of type H n with corresponding sets of helices H P and H P ′ defined by systems of the form ( i P , j P , P A , P B ) and ( f P ′ , g P ′ , P ′ A ′ , P ′ B ′ ) , respectively. Then | H P | = h Pn = | H P ′ | = h P ′ n . Proof.
Firstly, we suppose without loss of generality that, i P = f P ′ and j P = g P ′ .Then, we note that each helix h ∈ ( i P , j P , P A , P B ) gives rise to a unique helix h ′ ∈ ( i P , g P , P A ′′ , P B ′′ ), where P A ′′ and P B ′′ are suitable sets of pivoting entries in P . Theprocess consists of copying helix h , in such a way that each occurrence of entries of j P is substituted by a corresponding occurrence of g P (taking into account the new setsof pivoting vertices, P A ′′ and P B ′′ ), conversely, each occurrence of g P is substitutedby a corresponding occurrence of j P , keeping without changes the remaining rowsvisited by the helix h . For example, if a vertical arrow v ∈ h connects entries p i,j (starting vertex) and p i ′ ,j (ending vertex) in P then the corresponding vertical arrow v ′ ∈ h ′ connects entries of the rows i and i ′ if i ∈ { j P , g P } and i ′ / ∈ { j P , g P } or if i and i ′ are such that i, i ′ / ∈ { j P , g P } , v ′ connects rows i and j ( g ) if i ′ = g ( i ′ = j ).We let σ denote the bijection, σ : ( i P , j P , P A , P B ) → ( i P , g P , P A ′′ , P B ′′ ) defined bythese substitutions. Thus, if a bijection δ : ( i P , g P , P A ′′ , P B ′′ ) → ( f P , g P , P A ′ , P B ′ ) isdefined as σ where P A ′ and P B ′ are sets of pivoting entries of P given by P ′ A ′ and P ′ B ′ respectively, then the maps composition δσ is also a bijection from ( i P , j P , P A , P B ) to( f P , g P , P A ′ , P B ′ ). Any helix h ′ ∈ ( f P , g P , P A ′ , P B ′ ) corresponds uniquely to an helix h ′′ ∈ ( f P ′ , g P ′ , P ′ A ′ , P ′ B ′ ) via the identification τ : P → P ′ such that τ ( p i,j ) = p ′ i,j ,in this case p i,j ∈ P A ′ ( p i,j ∈ P B ′ ) if and only if p ′ i,j ∈ P ′ A ′ ( p ′ i,j ∈ P ′ B ′ ). In general,a copy h ′′ ∈ ( f P ′ , g P ′ , P ′ A ′ , P ′ B ′ ) of an helix h ′ ∈ ( f P , g P , P A , P B ) can be built takinginto account that an initial exterior vertex e f,j ∈ h ′ has a vertex e ′ f,j ′ ∈ f A ′ as itscorresponding initial exterior copy and h ′′ visits the same rows in the same order asthose visited previously by h ′ . We are done. (cid:3) Examples of copies of elements of the set(4 P , P , P A = { p , , p , , p , } , P B = { p , , p , , p , } ) A.M. Ca˜nadas et al associated to a matrix P of type H and defined by the word W P = BAABAB aregiven in section 6 (see figures ((55), (57), (59), (61)). In such a case, the correspondingcopies belong to the set(4 P , P , P A = { p , , p , , p , } , P B = { p , , p , , p , } )and the matrix P is partitioned according to the word ABABBA . Corollary 8.
Let W P = l m . . . l m n and W ′ P = l ′ m . . . l ′ m n ; l m n , l ′ m n ∈ { A, B } be two words associated to a matrix P of type H n with corresponding sets of helices H P = ( i P , j P , P A , P B ) , and H ′ P = ( f ′ P , g ′ P , P ′ A , P ′ B ) . Then | H P | = | H ′ P | . Proof.
The result follows from Theorem 7 by replacing, P ′ , A ′ , B ′ , P ′ A ′ , P ′ B ′ , f P ′ and g P ′ for P, A, B, P ′ A , P ′ B , f ′ P and g ′ P , respectively. (cid:3) Corollary 9.
If for n ≥ , P and P ′ are equivalent preprojective Kronecker moduleswith dimension vector of the form [ n + 1 n ] and corresponding sets of helices H P and H P ′ then | H P | = | H P ′ | . Proof.
Matrix presentations of preprojective Kronecker modules P and P ′ are bothof type H n defined by words of the form AA . . . ABB . . . BB . (cid:3) Theorem 10.
If for n ≥ , P denotes a preprojective Kronecker module then thenumber of helices associated to P is h Pn = n ! ⌈ n ⌉ where ⌈ x ⌉ denotes the smallest integergreatest than x . Proof.
According to Theorem 7 and Corollary 9, it suffices to determine the numberof helices | (1 p , ( n + 1) p , p A = { a i +1 ,i | ≤ i ≤ n } , p B = { b i,i | ≤ i ≤ n } ) | associatedto preprojective Kronecker modules p = ( n + 1 , n ) of type III and words of the form W p = AA . . . ABB . . . B .Firstly, we note that there is only one helix associated to the indecomposable prepro-jective modules (2 ,
1) and (3 , ,
3) with a ,j fixed are: hl = { a ,j , b , , b , , a , , a , , b , , b , , a , } ,hl = { a ,j , b , , b , , a , , a , , b , , b , , a , } ,hl = { a ,j , b , , b , , a , , a , , b , , b , , a , } ,hl = { a ,j , b , , b , , a , , a , , b , , b , , a , } . (33)The number of helices is given by the number of vertices at the last level of the followingassociated tree: rauer Configuration Algebras... ( a ,j , b , ) v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (cid:15) (cid:15) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ b , (cid:15) (cid:15) b , (cid:15) (cid:15) b , } } ④④④④④④④④④④④④ ! ! ❈❈❈❈❈❈❈❈❈❈❈❈ a , (cid:15) (cid:15) a , (cid:15) (cid:15) a , (cid:15) (cid:15) a , (cid:15) (cid:15) b , a , b , b , (34)Suppose now that the result is true for any indecomposable preprojective Kroneckermodule ( t + 1 , t ), 1 ≤ t < n then we can see that in general the rooted tree T n associated to the indecomposable preprojective Kronecker module ( n + 1 , n ) has thefollowing characteristics bearing in mind that vertex b , gives the root node a :( a ) a has n children enumerated from the left to the right as ( a , a , . . . , a n ),( b ) For 1 ≤ i ≤ n − a i has n − a i, , a i, , . . . , a i,n − ) whereas vertex a n has n − a n, , a n, , . . . , a n,n − ), each children of a vertex a n,l , 1 ≤ l ≤ n − n − a n,l ,l with 1 ≤ l ≤ n −
2, in general for this particular tree a vertex a n,l ,l ,l ,...,l t has n − ( t + 1) children, 1 ≤ t ≤ n −
2. Note that the number ofvertices at the last level of the rooted tree T ′ n with a n as root node is ( n − c ) For each h , 1 ≤ h ≤ n −
2, vertex a i,h is a root node of the tree T n − .The following picture shows the general structure of the rooted tree T n n -children( n − · · · ( n − · · · ( n − T ( n − · · · T ( n − · · · T ( n − ( n − · · · ( n − · · · ( n − n − .................................................................................................................................................... .................................................................................................... ........................................................................................................................................ ................................................................................................................................................. .................................................................................................... ......................................................................................................................... ............ ....................................................................................................................................... .................................................................................................... ........................................................................................................................... ............ ............................................................................................................................................................................................................................................................................................................ .................................................................................................... ........................................................................................................................................................................................................ (35) A.M. Ca˜nadas et al
According to the rules ( a ) − ( c ) the number of vertices L T n at the last level of the tree T n is given by the formula L T n = ( n − n − L T n − + L ( T ′ n ) = ( n − n − h pn − n − n − n − ⌈ n ⌉ = h pn n . (36)We are done. (cid:3) Remark 11.
Sequence A052558 is categorified via the number of helices associated topreprojective Kronecker modules, if in the condition ( e ) of its definition, it is assumedthat the starting vertex is fixed. Without such fixing condition the number of helicesassociated to a preprojective Kronecker module is given in the sequence encoded asA052591 in the OEIS. In this section, categorification of elements of the integer sequence A052591 are givenvia the number of summands in the heart of indecomposable projective modules overthe Brauer configuration algebra Λ K n defined by the Brauer configuration K n =( K n , K n , µ, O ) with the following properties for n ≥ K n = { x , x } ,K n = { V t = x (2 t +2)!1 x (( t )(2 t +2)!)2 } ≤ t ≤ n . (37)2. The orientation O is defined in such a way that for t ≥ x ; V (4!)1 < V (6!)2 < V (8!)3 < · · · < V ((2 n +2)!) n , at vertex x ; V < V < V < · · · < V ((( n )(2 n +2)!)) n . (38)3. The multiplicity function µ is such that µ ( x ) = µ ( x ) = 1.Where the symbol x ( j ) i in a given polygon V t means that occ( x i , V t ) = j . Note that, thespecializations x = 1 and x = 2 allows to describe polygons V t as integer partitionsof numbers in the sequence A052591 (see identity (15)).The following is the Brauer quiver Q K n associated to this configuration (numbers n t ( n t ) attached to the loops denote the occurrence of the vertex ( x above, x below) in the corresponding polygon V t , 1 ≤ t ≤ n ), c i t ( c j t ) denotes a set of loopsassociated to the vertex V t , | c i t | = l t = n t − | c j t | = l t = n t − rauer Configuration Algebras... V V V V Vn − Vn c i c i c i c i c i n [(2 n + 2)!] c j c j c j c j c j n [( n )(2 n + 2)!] ◦ ◦ ◦ ◦ . . . ◦ ◦ α α α α n β n +1 α n +1 β β β β n I is generated by the following relations (in this case, if there areassociated l t ( l t ) loops at the vertex V t associated to x (associated to x ) then welet P jt denote the product of j ≤ l mt loops, m ∈ { , } ), c mh s is a notation for a set ofcycles { c mh s, , c mh s, , . . . , c mh s,lms , m ∈ { , } , h ∈ { i, j } , s ∈ { , , . . . , n }} :1. c i s,x c i s,y − c i s,y c i s,x , for all possible values of i, s, x, y ,2. c j s,x c j s,y − c j s,y c j s,x , for all possible values of i, s, x, y ,3. c i s,x c j s,y and c j s,x c i s,y , for all possible values of i, s, x, y ,4. c i s,x β s +1 ; c j s,y α s +1 ; β s c i s,x ; α s c j s,x , for all possible values of i, s, x, y ,5. ( c i s,x ) ; ( c j s,y ) , for all possible values of i, s, x, y ,6. α t α t +1 ; α n +1 α ; β t β t +1 ; β n +1 β ; α t β t +1 ; β j α j +1 ; α n +1 β ; β n +1 α , forall possible values of j, t ,7. α i P ji γ i +1 ; α n +1 P j γ ; β t P ht γ t +1 ; β n +1 P h γ ; 0 < j < l i , 0 < h < l t , 1 ≤ i, t ≤ n , γ ∈ { α, β } ,8. For all the possible products (special cycles) of the form: ε = α t P l t t α t +1 P l t +1 t +1 . . . α n P l n n α n +1 P l . . . α t − P l t − t − ,ε = P mt − α t P l t t α t +1 P l t +1 t +1 . . . α n P l n n α n +1 P l . . . α t − P l t − − jt − ,ε = β t P l t t β t +1 P l t +1 t +1 . . . β n P l n n β n +1 P l . . . β t − P l t − t − ,ε = P ht − β t P l t t β t +1 P l t +1 t +1 . . . β n P l n n β n +1 P l . . . β t − P l t − − ht − , (40)relations of the form ε ri − ε sj , r, s ∈ { , , , } , i, j ∈ { , } take place. Notethat, products of the form P t − correspond to suitable orthogonal primitiveidempotents e t , 1 ≤ t ≤ n , A.M. Ca˜nadas et al ε α t , ε β t .The following result holds for indecomposable projective modules over the algebraΛ K n . Corollary 12.
For n ≥ fixed and ≤ t ≤ n , the number of summands in theheart of the indecomposable projective representation V t over the Brauer configurationalgebra Λ K n equals the number of helices associated to the preprojective Kroneckermodule (2 t + 3 , t + 2) , ≤ t ≤ n . Proof.
Firstly we note that for any t , rad V t = 0. Thus according to the Theorem 1the number of summands in the heart of any of the indecomposable projective modules V t equals occ( x , V t )+occ( x , V t ) = (2 t +2)!+ t (2 t +2)! = h p t +2 = h (2 t +3 , t +2)2 t +2 which isthe number of helices associated in a unique form to the indecomposable preprojectiveKronecker module (2 t + 3 , t + 2). We are done. (cid:3) The following results regard the dimension of algebras of type Λ K n . Corollary 13.
For n ≥ fixed, it holds that (dim k Λ K n ) = n + t γ n − + t δ n − , where γ n = n P m =1 m (2 m + 2)! , δ n = n P m =1 (2 m + 2)! , and t h denotes the h th triangular number. Proof.
Proposition 3 allows to conclude that dim k Λ K n /I = 2 n + P i =1 | C i | ( | C i | − i = 1 , | C i | = val ( x i ). The theorem holds taking into account thatfor any j ≥ j ( j −
1) = 2 t j − . (cid:3) Corollary 14.
For n ≥ fixed, it holds that dim k Z (Λ K n ) = n − n P t =1 h p t +2 . Proof.
Since rad Λ K n = 0, the result is a consequence of Theorem 4 with µ ( x ) = µ ( x ) = 1, | K n | = 2, | K n | = n and occ( x , V t ) + occ( x , V t ) = h p t +2 . (cid:3) Remark 15.
Similar results as in Corollaries 12-14 can be obtained for preprojectiveKronecker modules of the form (4 t + 2 , t + 1) , t ≥ K n = { x , x } ,K n = { V t = x (4 t +1)!1 x t (4 t +1)!)2 } ≤ t ≤ n , (41)and keeping the relations in the quiver without changes (bearing in mind of course thenew occurrences of the vertices for the different products). In particular, it holds thatdim k Z (Λ K n ) = n − n P t =1 h p t +1 . In this section elements of the integer sequence h n = n + ( n + 1) , n ≥ rauer Configuration Algebras... tetrad, such interpretation allows to categorify this integer sequence (encoded in theOEIS as A100705). Besides, some new Brauer configuration algebras are defined inorder to get alternative categorifications of this sequence and some additional integersequences. We note that the Brauer configuration (48) allows to see each polygon V n as a partition of the number h n into two parts of the form { n, n + 1 } where n occurs( n ) times and n + 1 occurs n + 1 times. Assuming the classical notation for partitions[1] each number h n can be expressed as follows:( n ) ( n ) ( n + 1) ( n +1) , n ≥ , (42)we let P n denote such a partition. The partition tree T P n associated to each partitionof the form P n is obtained by assuming the notation:1 / / /o/o/o • / / • / / /o/o/o • ~ ~ ⑦⑦⑦⑦ ❇❇❇❇ / / /o/o/o • ~ ~ ⑦⑦⑦⑦ (cid:15) (cid:15) ❇❇❇❇ ... ... ... ... (43)In this case, T P n has a root node with n + 1 children, n of them have n children andthe last one has n + 1 children in such a way that in the last level of T P n , n of thesechildren represent a partition of the form ( n ) ( n − ( n +1) (1) and the last one representsa partition of the form ( n ) ( n ) ( n + 1) (1) . Partition trees of the form T P n are used inthe proof of Theorem 18.The following results regard the number of cycles associated to matrices of type C n .In particular to preprojective representations of the tetrad of type IV.The next Theorem 16 can be proved by using similar arguments as those posed in theproof of Theorem 7. Theorem 16.
For n ≥ , let, ( U, U i , n | ≤ i ≤ , ( U ′ , U ′ i , n | ≤ i ≤ , C U and C U ′ be two matrices of type C n with corresponding sets of cycles C U and C U ′ defined bysystems of the form ( i RA , j RA , j ′ RA ′ , P A ri | ≤ i ≤ , r ∈ { , } ) and ( f RB , g RB , g ′ RB ′ , P B ri | ≤ i ≤ , r ∈ { , } ) , respectively. Then |C U | = |C U ′ | . Proof.
To each cycle
C ∈ ( i RA , j RA , j ′ RA ′ , P A ri | ≤ i ≤ , r ∈ { , } ), it is possible tobuild a unique copy C ′ ∈ ( f RB , g RB , g ′ RB ′ , P B ri | ≤ i ≤ , r ∈ { , } )) by using the sameprocedures described in the proof of Theorem 7. (cid:3) The next corollary shows that the number of cycles associated to an indecomposablerepresentation of type IV of the tetrad is invariant under admissible matrix transfor-mations. A.M. Ca˜nadas et al
Corollary 17.
If for n ≥ , U and U ′ are equivalent preprojective representations ofthe tetrad of type IV with corresponding sets of cycles C U and C U ′ then |C U | = |C U ′ | . Proof.
The matrix presentations of preprojective representations of type IV of thetetrad are of type C n . (cid:3) Theorem 18.
For n ≥ fixed and ≤ i ≤ n the number of cycles associated toan indecomposable preprojective representation of type IV and order i + 1 is h i = i + ( i + 1) . Proof.
Fix a preprojective representation U n of type IV and order n ≥
2, and denoteits different blocks as follows: U n = A B C DA ′ B ′ C ′ D ′ We note that all the cycles associated to U n can be seen as trees T c ( n +1) , ( n +1) whichhave the entry c ( n +1) , ( n +1) as root node with n branches whose successors are givenby entries c ′ , ( n +1) , c ′ , ( n +1) , . . . , c ′ n, ( n +1) .Each entry c ′ i, ( n +1) has n − i = n , whereas c ′ n, ( n +1) has n branches.Besides, all of these entries give rise to an arrow c ′ i, ( n +1) → d j,i ,for some entry d j,i ∈ D . Actually, d j,i is a successor root of c ′ i, ( n +1) with ( n − j ∈ { , . . . , n } and i = 1. If i = 1 then d ,j hasby construction n branches in C ′ . Therefore, the structure of T c ( n +1) , ( n +1) has thefollowing shape: c ( n +1) , ( n +1) c ′ , ( n +1) · · · c ′ i, ( n +1) · · · c ′ n, ( n +1) d n, · · · d i, · · · d , d n,n · · · d i,n · · · d ,n ............................................................................................................................................ .................................................................................................... ............................................................................................................................... ................................................................................................................................................ .................................................................................................... ........................................................................................................................ ............ .................................................................................................................................... .................................................................................................... ........................................................................................................................ ............ (44)Which corresponds to the partition tree T P ( n − of h ( n − = ( n − + ( n ) . (cid:3) As an example the following is the diagram of T c , such that the number of vertices inthe last level gives the number of associated cycles (described in the proof of Theorem18) to the indecomposable representation of the tetrad U : rauer Configuration Algebras... c c ′ c ′ c ′ d d d d d d d • • • • • • •• • • • • • • • • • • ........................................................................................................................................................................................................................................................................................................... ..................................................................................................... ............................................................................................................................................................................................................................................................................................... ................................................................ ........................................ ............ .................................................... ........................................ ............ ......................................................................... ...................................... ............................................................. ............................................................................... ................................................................... ................................................................... ........................................................................................................................................ ................................................................... ................................................................... ................................................................... ........................................................................................................................................ ................................................................... ................................................................... ................................................................... ................................................................... ................................................................... ........................................................................................................................................ The number of cycles associated to the indecomposable preprojective representation ofthe tetrad U (shown below) equals the second term of the integer sequence A100705.Actually, the number of cycles associated to U n is given by h ( n − = ( n − + ( n ) , n ≥
2, which is the ( n − U = ❄ ✲✻ ✲ ❄ ✲✻✛❄✛✻✛ Partition trees T P j associated to numbers h j = j + ( j + 1) in the proof of Theorem18 define a sequence of Brauer configuration algebras Λ E j , j ≥ E j whose set of vertices are 4-vertex paths contained in such trees. Inorder to describe Brauer configurations E j , we assume the notation L i i ,i +1 for the i th, 4-vertex path occurring in a third ramification of size i in the partition tree T P i .In fact, vertices in E j is a labeling of 4-vertex paths in partition trees T P i , 1 ≤ i ≤ j .As an example, the five 4-vertex paths of T P belong to P = P ∪ P ∪ P , where P = { L , } , P = { L , , L , , L , , L , } , and P = ∅ . The seventeen 4-vertex paths P of T P = T c , are given by the following identities: P = P ∪ P ∪ P ,P = { L , , L , , L , , L , } ,P = { L , , L , , L , , L , , L , , L , , L , , L , , L , } . (45)For j ≥ ≤ h i ≤ i , i + 1 ≤ h ′ i ≤ i , and 1 ≤ h i +1 ≤ ( i + 1) . The Brauerconfiguration E j = ( E j , E j , µ j , O j ) is defined in the following fashion: A.M. Ca˜nadas et al E j = { L ih i ,i , L ih ′ i ,i +1 , L i +1 h i +1 ,i +1 | ≤ i ≤ j } ∪ { L , } ,E j = { T P i | ≤ i ≤ j } ,T P i = P i = P ii ∪ P ii +1 ∪ P i +1 i +1 ,P ii = { L i ,i , L i ,i , . . . , L ii ,i } , for i ≥ ,P ii +1 = { L ii +1 ,i +1 , L ii +2 ,i +1 , . . . , L ii ,i +1 } , for i ≥ ,P i +1 i +1 = { L i +11 ,i +1 , L i +12 ,i +1 , . . . , L i +1( i +1) ,i +1 } , for i ≥ . (46)2. The orientation O j for successor sequences is defined by the usual order ofnatural numbers, i.e., any successor sequence has the form T ( s j ) P < T ( s j ) P < · · · < T ( s j ( j − ) P j − < T ( s jj ) P j , for some nonnegative integers s j m ( s j m = 0, meansthat the vertex does not occur in the polygon T P m ).3. µ j is a multiplicity function such that, µ j ( L ) = 2 for any L ∈ { L , } ∪ P ii +1 ∪ P j +1 j +1 , 2 ≤ i ≤ j , µ j ( L ) = 1, otherwise. This multiplicity function is defined toavoid the presence of truncated vertices in the configuration.The Brauer quiver Q E j has the following shape, where notation c L ihr,s means that thecorresponding polygon has associated loops of type c L ihr,i +1 , c L j +1 hm,j +1 , i +1 ≤ h r ≤ i ,1 ≤ h m ≤ ( i + 1) defined by vertices L ih r ,s ( h = 1). In this case, α L iir,s ( β L iir,s )denotes arrows determined by polygons T P ( i − and T P i , 1 ≤ i r ≤ i , i ≥
2. Loops c L h , in the diagram appear if j = 5. If j > T P there areassociated only loops of type c L h , and in the vertex T P j there are attached loops ofthe form c L jhj,j and c L j +1 hj +1 ,j +1 . ◦ T P ◦ T P ◦ T P ◦ T P ◦ T P . . .α L i , α L i , α L i , α L i , β L i , β L i , β L i , β L i , c L h , c L h , c L h , c L h , c L h , c L h , (47)The ideal J in this case is generated by the following set of relations defined for allthe possible values of h, i, j, m, t, t ′ and u :1. ( c L ij,m ) ,2. c L jhj,j c L j +1 h ( j +1) ,j +1 ,3. c L ihj,m α L i +1 ij,m +1 ,4. α L iij,m c L ih ( j +1) ,m ,5. α L ji ( j − ,j c L j +1 h ( j +1) ,j +1 , rauer Configuration Algebras... c L ihj,m β L ii ( j − ,m ,7. c L j +1 h ( j +1) ,j +1 β L ji ( j − ,j ,8. β L iij,m c L i − hj,m − ,9. α L iij,m α L i +1 i ( j +1) ,m +1 ,10. β L iij,m β L i − i ( j − ,m − ,11. s L it,u a , where a is the first arrow of the special cycle s L it,u associated to thevertex L it,u ,12. s L it,i − s L it ′ ,i , s L it,i − s L ih,i +1 , s L it,i − s L i +1 m,i +1 .If we let Λ E j denote the algebra kQ E j /J , then the following result categorifies sequence h n = n + ( n + 1) , n ≥
1, by considering its elements as invariants of objects of thecategory mod Λ E j . Theorem 19.
For j ≥ fixed and ≤ i ≤ j , the number of summands in the heart ofthe indecomposable projective module T P i over the Brauer configuration algebra Λ E j equals the number of cycles associated to a preprojective representation of the tetradof type IV and order i + 1 . Proof.
Since for any indecomposable projective module T P i , it holds that rad T P i = 0then the theorem follows from Theorem 1 and the definition of the polygon T P i whichhas i + ( i + 1) nontruncated vertices counting repetitions. (cid:3) Numbers in the sequence A100705 appear by computing the dimension of quotientspaces of the form Λ E i +1 /F i +1 , where F i +1 is a k -subspace of Λ E i +1 isomorphic toΛ E i . Actually the following results hold. Theorem 20.
For j ≥ fixed, it holds that dim k Λ E j = ( j +1)( j +2)(2 j +3)6 + ( j ( j +1)2 ) +2 j − . And dim k Λ E n +1 − dim k Λ E n = h n +1 + 2 , where h n = n + ( n + 1) , n ≥ . Proof.
Note that, | E j | = j , and E j = H ∪ H , where H = { L ih i,i | ≤ i ≤ j + 1 } and H = { L mh m,m +1 | ≤ m ≤ j } , L h , = L , , 1 ≤ h i ≤ i , 1 ≤ h m ≤ m − m (terms L mh s,m +1 correspond bijectively to vertices L mm + s,m +1 ∈ P mm +1 ). If x ∈ H then val ( x ) = 2 and µ j ( x ) = 1, whereas, if y ∈ H then val ( y ) = 1 and µ j ( y ) = 2. Theresult follows bearing in mind that val ( L h , ) = 1, µ j ( L h , ) = 2 and that for i and m fixed, there are i vertices of type H , 1 ≤ i ≤ j + 1 and m − m vertices of type H , 2 ≤ m ≤ j . We are done. (cid:3) Theorem 21.
For j ≥
3, dim k Z (Λ E j ) − dim k Z (Λ E ( j − ) = j − j + 1. Proof.
Since rad Λ E j = 0, | E j | = j P i =1 i + ( j + 1) and the number of loops in thequiver Q E j equals | C E j | then dim k Z (Λ E j ) = j P i =1 ( i − i ) + ( j + 2). We are done. (cid:3) The sequence j − j + 1 appears encoded in the OEIS as A100104. A.M. Ca˜nadas et al
In this section, we describe the way that some Brauer configuration algebras can bedefined by using integer sequences.
We consider Brauer configuration algebras of the form Λ Γ n = kQ Γ n /J induced bythe Brauer configuration Γ n whose polygons are defined by integer partitions of theelements in the sequence A100705 (see 42). And such that For n ≥ n =(Γ , Γ , µ, O ) with1. Γ = { , , , . . . , n, n + 1 } , Γ = { V t = t ( t ) ( t + 1) ( t +1) } ≤ t ≤ n , i.e., occ( t, V t ) = t , occ( t + 1 , V t ) = t + 1 . (48)2. The orientation O is defined in such a way that for 2 ≤ i ≤ n at vertex i , V ( i,< ) i − < V ( i ,< ) i , where V ( y,< ) x means that the polygon V x occurs y times inthe successor sequence of the corresponding vertex, in particular, V i − < V i . Atthe vertex n + 1, the successor sequence has the form V ( n +1 ,< ) n , in this case, V n, < V n, < · · · < V n,n < V n,n +1 where V n,i denotes the i th occurrence of thepolygon V n in the sequence.3. The multiplicity function µ is such that µ ( j ) = 1, for any j ∈ Γ .The following is the quiver Q Γ n associated to the Brauer configuration Γ n , worthnoting that there is no arrow connecting vertex 1 with any other vertex provided thatit is truncated (see Theorem 1, item 5), besides we use the symbol [ x j ; y j ] to denotethat the vertex x j occurs y j times at the polygon h j = j + ( j + 1) (see identity (15)).And c ij is a set of loops { c ij y | ≤ y ≤ occ( x j , h j ) − , ≤ i ≤ n + 1 } . For instance, at17 there are associated the loops, c , c , c and c , c . c c c c c c c c c ◦ ◦ ◦ ◦ ◦ . . .α α α α β β β β [2; 2] [3; 3] [4; 4] [5; 5] [6; 6][2; 4] [3; 9] [4; 16] [5; 25] (49) rauer Configuration Algebras... The following are examples of polygons in a Brauer configuration Γ n :5 = (1) + (2 + 2) = (1) (1) (2) (2) ,
17 = (2 + 3) + (2 + 3) + (2 + 2 + 3) = (2) (4) (3) (3) ,
43 = (3 + 3 + 4) + (3 + 3 + 4) + (3 + 3 + 4) + (3 + 3 + 3 + 4) = (3) (9) (4) ,
89 = (4 + 4 + 4 + 5) + (4 + 4 + 4 + 5) + (4 + 4 + 4 + 5) + (4 + 4 + 4 + 5) + (4 + 4 + 4 + 4 + 5)... = ... (50)The ideal J is generated by the following relations where for a fixed 2 ≤ l ≤ n + 1, P i,lh j is the product of i loops of type l (1 ≤ i ≤ occ( l, h j ) −
1) attached to the polygon h j with y j − y j ∈ { j , j } ):1. c uj x c vj y , if u = v , for all the possible values of u , v , x , y and j ,2. c tj x c tj y = c tj y c tj x , for all the possible values of x , y , t , and j ,3. ( c tj x ) for all the possible values of j , t and x ,4. c hj x α h +1 ; α h c h +1( j +1) x ; c hj x β h − ; β h c h − j − x , α j β j for all the possible values of h, j and x ,5. α i α i +1 ; β j +1 β j , 2 ≤ i ≤ n −
1, 2 ≤ j ≤ n − ε j = P u,jh j α j P y j +1 − ,jh j +1 β j P y j − (1+ u ) ,jh j ,ε j = α j P y j +1 − ,jh j +1 β j P y j − ,jh j ,ε j = P u,jh j +1 β j P y j − ,jh j α j P y j +1 − (1+ u ) ,jh j +1 ,ε j = β j P y j − ,jh j α j P y j +1 − ,jh j +1 ,ε j +1 = P v,j +1 h j +1 α j +1 P y j +2 − ,j +1 h j +2 β j +1 P y j +1 − (1+ v ) ,j +1 h j +1 ,ε j +1 = α j +1 P y j +2 − ,j +1 h j +2 β j +1 P y j +1 − ,j +1 h j +1 ,ε j +1 = P v, +1 h j +2 β j +1 P y j +1 − ,j +1 h j +1 α j +1 P y j +2 − (1+ v ) ,j +1 h j +2 ,ε j +1 = β j +1 P y j +1 − ,j +1 h j +1 α j +1 P y j +2 − ,j +1 h j +2 , (51)then there are relations of the form ε rs − ε r ′ s ′ where r, r ′ ∈ { , . . . , } , r = r ′ and s, s ′ ∈ { j, j + 1 } , for all the possible values of u , v and j ,7. ε j α j , ε j β j , ε j +1 α j +1 , ε j β j +1 .The following results are consequences of Theorems 1, 4, and Proposition 3. Corollary 22.
For n ≥ fixed and ≤ i ≤ n , the number of summands in the heartof the indecomposable projective module V i over the algebra Λ Γ n is i + i + 1 . Proof.
Since for any indecomposable projective module V i , it holds that rad V i = 0then the theorem follows from Theorem 1 and the definition of the polygon V i whichhas i + i + 1 nontruncated vertices counting repetitions. (cid:3) . A.M. Ca˜nadas et al
Corollary 23.
For n ≥ fixed, dim k Λ Γ n = n P m =2 ( m ( m +1)) − ( n − n +1)( n +2) .And dim k Λ Γ n +1 /G n +1 = 2(1 − t n ) + [( n + 1)( n + 2)] , where for i ≥ , t i denotes the i th triangular number. And G n +1 is a k -subspace of Λ Γ n +1 isomorphic to Λ Γ n Proof.
It is enough to observe that for n ≥ ≤ j < n + 1, it holds that val ( j ) = j + j , whereas val (1) = 1 and val ( n + 1) = n + 1. The theorem holds as aconsequence of Proposition 3. (cid:3) Corollary 24.
For n ≥ fixed, it holds that dim k Z (Λ Γ n ) = n ( n +1)( n +2)3 + 1 . And dim k Z (Λ Γ n +1 ) /Z n +1 = 2 t n +1 , where Z n +1 is a k -subspace of Z (Λ Γ n +1 ) isomorphicto Z (Λ Γ n ) . Proof.
Since rad Λ Γ n = 0, the result is a consequence of Theorem 4 with µ ( i ) = 1,for any 2 ≤ i ≤ n + 1, | Γ | = n , | Γ | = n , occ( i, h i ) + occ( i + 1 , h i ) = i + i + 1,2 ≤ i ≤ n , and occ(2 , h ) = 2. (cid:3) Remark 25.
Note that Corollaries 22-24 are categorifications of the integer sequences n + n + 1 (encoded in the OEIS as A002061), n P m =2 ( m ( m + 1)) − ( n − n + 1)( n +2), and n ( n +1)( n +2)3 + 1 (which is the sequence A064999). Elements of the sequenceA064999 appear as coefficients (in the case t = 3) of the generating polynomial ofa n -twist knot with the form P n ( x ) = P t ≥ a n,t x t . And the sequence n P t =2 ( t ( t + 1)) = P ≤ i 0) to( n, n ) with steps (1 , 0) and (0 , x = y ofthe plane ( x, y )-plane. It can be shown that e ( P n ) is given by the n th Catalan number C n = n +1 (cid:0) nn (cid:1) .We define now a family of Brauer configuration algebras Λ D n , n > D n whose nontruncated vertices are in correspondence with objects oftype D n , polygons are obtained by choosing objects of type D s , for 1 ≤ s ≤ n . Weassume the notation L sj,n ∈ D s for the j th object of type s in a given polygon. Withoutloss of generality, we assume that for the first polygon P , it holds that | P | = u > n ≥ D n = ( D n , D n , µ n , O n ) rauer Configuration Algebras... goes as follows: D n = { L si s ,n , ≤ s ≤ n, ≤ i s ≤ u s − u ( s − } ∪ P ,P = { L i ,n | ≤ i ≤ u } ,D n = { P h | ≤ h ≤ n } ,P h = P h ( h − ∪ P hh , | P h ( h − | = | P ( h − | , ≤ h ≤ n,P h ( h − = { L si s ,n | ≤ s ≤ h − } ,P hh = { L hi h ,n | ≤ i h ≤ u h − u ( h − , ≤ h ≤ n } ,µ n ( L ) = 1 , for any vertex L ∈ ( D ) n \ P nn ,µ n ( L ) = 2 , for any vertex L ∈ P nn , (52)In P h ( h − , it holds that, 1 ≤ i ≤ u if s = 1, and 1 ≤ i s ≤ u s − u ( s − , if s > O n is defined by the usual order of natural numbers. Thus, for avertex L ij,n ∈ D n \ P nn , the successor sequence has the form P i < P ( i +1) < · · · < P ( n − < P n For vertices L ( n − r,n , the successor sequence has the form P ( n − < P n , whereas forvertices of the form L nr,n ∈ P nn , the orientation is of the form P n < P n .The following is the shape of the Brauer quiver Q D n defined by D n . P P P ( n − P ( n − P j P P P n α α α α ( n − ) β β β β βjβ ( n − α ( n − β ( n − cLnin,n (46) (53) α j , β j and c L nin ,n denote j × A.M. Ca˜nadas et al α j = α jL i ,n α jL i ,n ... α jL sis ,n ... α jL jij ,n , β j = β jL i ,n β jL i ,n ... β jL sis ,n ... β jL jij ,n , c L nin ,n = c L n ,n c L n ,n ... c L ni ,n ... c L n ( un − u ( n − ,n where α jL sis ,n ( β jL sis ,n ) is a set of arrows defined by the successor sequence at vertex L si s ,n connecting the corresponding polygons (polygon P n with the corresponding P j ).And c L nin,n is a set of loops defined by vertices L ni n ,n , 1 ≤ i n ≤ u n − u ( n − .The following relations generate the admissible ideal J of the Brauer configurationalgebra Λ D n = kQ D n /J , for all possible values of i, i ′ , j, j ′ , r, r ′ and n .1. ( c L nr,n ) , c L nr,n c L nr ′ ,n , r = r ′ ,2. α jL ir ,n α jL i ′ r ′ ,n , i = i ′ ,3. α jL ir ,n α ( j +1) L i ′ r ′ ,n ,4. c L nin,n β i ,5. α ( n − c L nin,n ,6. For 1 ≤ j ≤ n , fixed and 1 ≤ i ≤ j , s L ij,n − s L jj ′ ,n where s x is a special cycleassociated to the vertex x ,7. β i α i ,products of the form; β i α i , α ( n − c L nin,n , c L nin,n β i means that relations of the form xx ′ , y ′ y and z ′ z have place where x ′ , y ′ and z ′ are entries of the corresponding matrices.The following result categorifies numbers of a counting function u t , for t ≥ Theorem 26. For ≤ i ≤ n and n > fixed, the number of summands in the heartof the indecomposable projective module P i over the algebra Λ D n is u i . Proof. Since rad P i = 0, then the number of summands in the heart of the indecom-posable projective module P i equals the number of its nontruncated vertices countingrepetitions, which by definition is given by the sum u + ( u − u ) + · · · + ( u ( i − − u ( i − ) + ( u i − u ( i − ) = u i . We are done. (cid:3) Theorem 27. For n ≥ fixed, dim k Λ D n = 2 n + n ( n − u + 2 n P i =2 t ( n − i ) ( u i − u ( i − ) . Proof. It suffices to note that val ( L ii s ,n ) = n − i + 1 and µ n ( L ii s ,n ) = 1 for any L ii s ,n ∈ D n \ P nn , whereas for any x ∈ P nn , it holds that val ( x ) = 1 and µ n ( x ) = 2. Weare done. (cid:3) Since for any n > 1, rad Λ D n = 0, then we have the following result regarding thecenter of these algebras. rauer Configuration Algebras... Theorem 28. For n ≥ fixed, dim k Z (Λ D n ) = ( u n − u ( n − ) + ( n + 1) . Proof. Note that | D n | = u n , | D n | = n , P α ∈ D n µ n ( α ) = 2 u n − u ( n − . Since Loops Q D ( n − ) = | C D n | , the theorem holds. (cid:3) Remark 29. Perhaps, the sequence C n of Catalan numbers is one of the most in-teresting counting functions, they count the number of plane binary trees with n + 1endpoints (or 2 n + 1 vertices), the number of triangulations of an ( n + 3) polygon, orthe number of paths L in the ( x, y )-plane from (0 , 0) to (2 n, 0) with steps (1 , 1) and(1 , − 1) that never pass below the x -axis, such paths are called Dyck paths [18]. Thus,if u n = C n +1 , n ≥ C n viathese enumeration problems.Since the number of compositions (partitions in which the order of the summands isconsidered) of a positive integer n in which no 1’s appear is the Fibonacci number f ( n − [1]. Then Theorem 26 categorifies these numbers by assuming that u n = f ( n − with n ≥ 4. If j > 4, then dim k ( Z (Λ D j ) /C j ) − f ( j − , where C j is a k -subspaceof Z (Λ D j ) isomorphic to Z (Λ D j − ).Theorem 26 categorifies the sequence p ( n ) which gives the number of partitions of apositive integer n , recall the Hardy-Ramanujan theorem which states that for large n , p ( n ) ∼ n √ e π √ n (see also the sequence A002865 whose numbers give the differences p ( n ) − p ( n − u n = p ( n + 1), n ≥ M ( n ) encodedin the OEIS as A000372, which consists of Dedekind numbers, these numbers count thenumber of antichains in the powerset 2 n (i.e., the set consisting of all the subsets of n = { , , , . . . , n } ) ordered by inclusion or the number of elements in a free distributivelattice on n generators. In this case u n = M ( n ), for n ≥ 1, worth noting that up todate only 8 numbers of this sequence are known. References [1] G. Andrews, The Theory of Partitions , Cambridge University. Press, Cambridge, 1998.[2] S. Brenner, Endomorphism algebras of vector spaces with distinguished sets of subspaces ,J. Algebra (1967), 100-114.[3] , On four subspaces of a vector space , J. Algebra (1974), 587-599.[4] A. M. Ca˜nadas, I.D. Marin, and P.F.F. Espinosa, Categorification of some integer se-quences via Kronecker modules , JPANTA (2016), no. 4, 339-347.[5] P. Fahr and C. M. Ringel, A partition formula for Fibonacci numbers , J. Integer Seq. (2008), no. 08.14.[6] , Categorification of the Fibonacci numbers using representations of quivers , J.Integer Seq. (2012), no. 12.2.1.[7] , The Fibonacci triangles , Advances in Mathematics. (2012), 2513–2535.[8] I.M. Gelfand and V.A. Ponomarev, Problems of linear algebra and classification ofquadruples of subspaces in a finite dimensional vector space , Colloq. Math. Soc. J ˜A¡nosBolyai, Hilbert Space Operators, Tihany (1970), 163–237. A.M. Ca˜nadas et al [9] E.L. Green and S. Schroll, Brauer configuration algebras: A generalization of Brauergraph algebras , Bull. Sci. Math. (2017), 539–572.[10] D. Knuth, The Art of Computer Programming , Vol. 4, Addison-Wesley, 2004. Fascicle.3.[11] R.B. Lin, On the applications of partition diagrams for integer partitioning , Proc. The23rd workshop on combinatorial mathematics and computation theory (2006).[12] P. Luschny, Counting with partitions Representations of a tetrad , Izv. AN SSSR Ser. Mat. (1967), no. 4,1361-1378 (in Russian). English transl. in: Math. USSR Izvestija 1 (1967) 1305-1321,1969.[14] C.M. Ringel, Catalan combinatorics of the hereditary artin algebras . In Developments inRepresentation Theory, Contemp Math, 673, AMS, Providence, RI, 2016, 51-177.[15] S. Schroll, Brauer Graph Algebras , Springer, Cham, 2018. In: Assem I., Trepode S.(eds), Homological Methods, Representation Theory, and Cluster Algebras, CRM ShortCourses, 177-223.[16] A. Sierra, The dimension of the center of a Brauer configuration algebra , J. Algebra (2018), 289-318.[17] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Cate-gories , Gordon and Breach, London, London, 1992.[18] R. Stanley, Enumerative Combinatorics , Vol. 1, Cambridge University Press, Cambridge,1997.[19] A.G. Zavadskij and G. Medina, The four subspace problem; An elementary solution ,Linear Algebra Appl. (2004), 11-23.[20] A.G. Zavadskij, On the Kronecker problem and related problems of linear algebra , LinearAlgebra Appl. (2007), 26-62.[21] N.J.A. Sloane, On-Line Encyclopedia of Integer Sequences , Vol. http://oeis.org/A132262,A110122, A002061, A005586, A024166, A100705, A052558, A052591, The OEIS Founda-tion. Agust´ın Moreno Ca˜nadas Pedro Fernando Fern´andez [email protected] pff[email protected] of Mathematics Department of MathematicsUniversidad Nacional de Colombia Universidad Nacional de Colombia.Kra 30 No 45-03ZIP Code 11001000Bogot´a-ColombiaIsa´ıas David Mar´ın Gaviria Gabriel Bravo [email protected] [email protected] of Mathematics Department of MathematicsUniversidad Nacional de Colombia Universidad Nacional de Colombia.