In this section, the number of suitable cycles associated with indecomposable representations of the tetrad is related to combinatorial information arising from the indecomposable projective modules over some Brauer configuration algebras to categorify integer sequences.
3.1. Cycles Associated with a Preprojective Representation of Type IV
For , we consider -matrixes of type , whose rows and columns are partitioned by four adjacent matrix blocks (from the left to the right), and denoted , respectively. and are matrixes vertically adjacent (blocks of type as well as those of type are horizontally adjacent) of the same size, three of the four blocks consists of -matrixes, and the remaining block consists of two -matrixes.
If
U is a matrix of type
then
ith row (
jth column)
(
) of
U is given by the union
where
and
are corresponding rows in the matrixes
and
,
.
The
jth column
of
U is given by the union
The following is a typical shape of a matrix
U of type
:
Matrixes of type have associated cycles which are connected by an oriented graph whose construction is as follows:
- 1.
(Vertices) By definition any matrix U of type is defined by a word w whose letters are the symbols and , i.e., where is a permutation of four elements. For the sake of clarity, later on, we assume that the matrix blocks of a matrix U of type are organized according to with , . Blocks and are said to be external blocks, whereas blocks and are internal blocks.
- 2.
Fix three rows , and . In this case, is a -matrix block, whereas is a -matrix block. Thus, according to our choice, and .
- 3.
Entries of matrix U of type are either pivoting or exterior vertices (or entries). We consider rows and , which do not contain pivoting entries. If denotes the set of pivoting entries of the matrix , , with , . Then , if and .
- 4.
Without loss of generality, we assume that the starting and ending vertex are the same and belong to , it is a pivoting vertex. Furthermore, and .
- 5.
According to the assumption introduced in 1, the sequence of vertices connected by a cycle
is of the form,
where entries
,
,
and
correspond, respectively, to the blocks
,
,
and
.
- 6.
Any cycle contains the fixed vertices , , , and , which is a pivoting entry. Entries of the form . No horizontal arrow has an entry as its ending vertex.
- 7.
Vertices , , , , and are pivoting entries.
- 8.
(Arrows) arrows in connect alternatively pivoting entries with exterior entries. They are defined as follows:
- (a)
Arrows in are either horizontal (→, ←) or vertical (↑, ↓). We let denote the set of arrows of . We write, , , and , the different ways of connecting matrixes of the different blocks . According to the sequence of vertices, the first vertical arrow of the cycle is of the form .
- (b)
Since we are assuming cycles associated with a word of the form , therefore horizontal arrows connect adjacent matrixes () with () (conversely, () with ()). The first horizontal arrow in our case is of the form . Vertical arrows connect matrixes in the form or . External blocks are connected by a unique vertical arrow (either iff or iff ). Two arrows, , connect internal matrix blocks ( iff , iff ). Two vertical arrows connect internal matrix blocks or ; the same conditions satisfy matrixes and .
- (c)
Each horizontal arrow is preceded by a unique vertical arrow, and unless the first arrow, any vertical arrow is preceded by an horizontal arrow.
- (d)
No row or column of a matrix in a matrix block , , is visited by arrows of more than once.
Please note that any matrix U of type can be described in the form . We let denote the set of all cycles associated with a matrix U of type and its corresponding cardinal.
In this work, we establish an identity between the number of cycles associated with preprojective representations of type IV of the tetrad and an integer number (in the sense of (
5)) defined by the indecomposable projective modules over some Brauer configuration algebras.
Figure 5 shows the canonical matrix presentation of such preprojective representations where
is an
identity matrix,
(
) is an
-matrix, obtained by adding a row of zeros at the top (at the bottom) of an
-identity matrix.
n is said to be the
order of the representation.
The following
Figure 6 gives an example of a cycle of type
associated with a preprojective representation of order
of the tetrad, black arrows connect fixed vertices of the corresponding cycles, whereas
(
) denote the set of diagonal (subdiagonal) entries used to define the pivoting vertices.
3.2. Categorification of the Sequence A100705 and Some Related Integer Sequences
In this section, elements of the integer sequence , are interpreted as the number of cycles associated with preprojective representations of the tetrad. Such interpretation allows categorifying this integer sequence (encoded in the OEIS as A100705). Moreover, some new Brauer configuration algebras are defined to obtain alternative categorifications of others types of integer sequences.
Firstly, we note that the Brauer configuration (
10) allows seeing each polygon
as a partition of the number
into two parts of the form
where
n occurs
times and
occurs
times. Assuming the classical notation for partitions [
16] each number
can be expressed as follows:
we let
denote such a partition. The partition tree
associated with each partition of the form
is obtained by assuming the notation given in
Figure 7.
In this case, has a root node with children, n of them have n children and the last one has children in such a way that in the last level of , n of these children represent a partition of the form and the last one represents a partition of the form . Partition trees of the form are used in the proof of Theorem 4.
Now we consider Brauer configuration algebras of the form induced by the Brauer configuration such that For fixed, with
- 1.
- 2.
The orientation is defined in such a way that for at vertexi, , where means that the polygon occurs y times in the successor sequence of the corresponding vertex, in particular, . At the vertex , the successor sequence has the form , in this case, where denotes the ith occurrence of the polygon in the sequence.
- 3.
The multiplicity function is such that , for any .
Figure 8 shows a part of the quiver
associated with the Brauer configuration
; it is worth noting that there is no arrow connecting vertex 1 with any other vertex provided that it is truncated (see Theorem 1, item 5), moreover, we use the symbol
to denote that the vertex
occurs
times at the polygon
(see identity (
5)). Furthermore,
is a set of loops
. For instance, at 17 there are associated the loops,
and
.
The following are examples of polygons in a Brauer configuration
:
The ideal J is generated by the following relations where for a fixed , is the product of i loops of type l () attached to the polygon with being the total number of such loops ():
- 1.
, if , for all the possible values of u, v, x, y and j,
- 2.
, for all the possible values of x, y, t, and j,
- 3.
for all the possible values of j, t and x,
- 4.
; ; ; , for all the possible values of and x,
- 5.
; , , ,
- 6.
If
then there are relations of the form
where
,
and
, for all the possible values of
u,
v and
j,
- 7.
, , , .
The following results prove that the number of cycles associated with preprojective representations of type IV are invariant under admissible transformations.
Theorem 3. For , let, , , and be two matrixes of type with corresponding sets of cycles and defined by systems of the form and , respectively. Then .
Proof. Any cycle
with set of vertices of the form (
8) corresponds uniquely to a cycle
, via the identification
such that
, for any vertex
, where
Entries , , and belong, respectively, to the blocks , , and of .
In this case, if and only if . In general, a copy of a cycle can be built taking into account that an initial exterior vertex has a vertex as its corresponding initial exterior copy and visits the same rows in the same order as those visited previously by . We are done. □
The next corollary shows that the number of cycles associated with an indecomposable representation of type IV of the tetrad is invariant under admissible matrix transformations.
Corollary 1. If for , U and are equivalent preprojective representations of the tetrad of type IV with corresponding sets of cycles and then .
Proof. Matrix presentations of preprojective representations of type IV of the tetrad are of type . □
The following theorem regards Brauer configuration algebra
and its connection with the number of cycles associated with a preprojective representation of the tetrad of type IV. Recall that the notation
(see (
5)) is adopted for the integer number associated with the polygon
V, in this case
V is interpreted as an integer partition of
.
Theorem 4. For fixed and the number associated with the the polygon (see Figure 5 and Formulas is the number of cycles associated with an indecomposable preprojective representation of type and order . Moreover, the identification of numbers with partitions of the form defines a bijection between indecomposable projective modules over the Brauer configuration algebra and preprojective representations of type of the tetrad. Proof. According to Theorem 1, in order to prove that there exists the required bijection, it suffices to find out the number of cycles associated with a given preprojective representation of type IV. To do that, we fix a representation
of this type of order
, and denote its different blocks as follows:
We note that all the cycles associated with
can be seen as trees
which have the entry
as root node with
n branches whose successors are given by entries
Each entry
has
branches if
, whereas
has
n branches. Moreover, all of these entries give rise to an arrow
for some entry
. Actually,
is a successor root of
with
branches in the tree whenever
and
. If
then
has by construction
n branches in
. Therefore, the structure of
has the shape given in
Figure 9.
Which corresponds to the partition tree
of
, thus the correspondence
is a bijection between indecomposable preprojective representations of type IV of the tetrad and polygons of the Brauer configuration (
10). □
As an example, the following
Figure 10 illustrates the diagram of
such that the number of vertices in the last level gives the number of associated cycles (described in the proof of Theorem 4) with the indecomposable representation of the tetrad
:
The number of cycles associated with the indecomposable preprojective representation of the tetrad
(shown in
Figure 11) equals the second term of the integer sequence A100705. Actually, the number of cycles associated with
is given by
,
, which is the
th term of this sequence. Black arrows denote the common part of all these cycles.
The following results are consequences of Theorems 1 and 2, and Proposition 1.
Corollary 2. For fixed and , the number of summands in the heart of the indecomposable projective module over the algebra is .
Proof. Since for any indecomposable projective module , it holds that then the result follows from Theorem 1 and the definition of the polygon which has nontruncated vertices counting repetitions. □
Corollary 3. For fixed, . Furthermore, , where for , denotes the ith triangular number. Furthermore, is a -subspace of isomorphic to .
Proof. It is enough to observe that for and , it holds that , whereas and . The theorem holds as a consequence of Proposition 1. □
Corollary 4. For fixed, it holds thatwhere is a -subspace of isomorphic to . Proof. Since , the result is a consequence of Theorem 2 with , for any , , , , , and . □
Remark 1. Note that Corollaries 2–4 are categorifications of the integer sequences (encoded in the OEIS as A002061), , and (which is the sequence A064999). Elements of the sequence A064999 appear as coefficients (in the case ) of the generating polynomial of a n-twist knot with the form . Furthermore, the sequence is encoded A024166 in the OEIS.
Partition trees
associated with numbers
in the proof of Theorem 4 define a sequence of Brauer configuration algebras
,
induced by Brauer configurations
whose set of vertices are 4-vertex paths contained in such trees. In order to describe Brauer configurations
, we assume the notation
for the
th, 4-vertex path occurring in a third ramification of size
in the partition tree
. In fact, vertices in
define a labeling of 4-vertex paths in partition trees
,
. As an example, the five 4-vertex paths of
belong to
, where
,
, and
. The seventeen 4-vertex paths
of
are given by the following identities:
For fixed, , , and . The Brauer configuration is defined in the following fashion:
- 2.
The orientation for successor sequences is defined by the usual order of natural numbers, i.e., any successor sequence has the form , for some nonnegative integers (, means that the vertex does not occur in the polygon ).
- 3.
is a multiplicity function such that, for any , , , otherwise. This multiplicity function is defined to avoid the presence of truncated vertices in the configuration.
Figure 12 shows the Brauer quiver
, where notation
means that the corresponding polygon has associated loops of type
,
,
,
defined by vertices
(
). In this case,
(
) denotes arrows determined by polygons
and
,
,
. Loops
in the diagram appear if
. If
then at the vertex
there are associated only loops of type
and in the vertex
there are attached loops of the form
and
.
The ideal J in this case is generated by the following set of relations defined for all the possible values of and u:
- 1.
,
- 2.
,
- 3.
,
- 4.
,
- 5.
,
- 6.
,
- 7.
,
- 8.
,
- 9.
,
- 10.
,
- 11.
, where a is the first arrow of the special cycle associated with the vertex ,
- 12.
, , .
If we let denote the algebra , then the following result categorifies sequence , , by considering its elements as invariants of objects of the category .
Theorem 5. For fixed and , the number of summands in the heart of the indecomposable projective module over the algebra is .
Proof. Since for any indecomposable projective module , it holds that then the theorem follows from Theorem 1 and the definition of the polygon which has nontruncated vertices counting repetitions. □
Numbers in the sequence A100705 appear by computing the dimension of quotient spaces of the form , where is a -subspace of isomorphic to . Actually the following results hold.
Theorem 6. For fixed, it holds that . Furthermore, , where , .
Proof. Please note that , and , where and , , , (terms correspond bijectively to vertices ). If then and , whereas, if then and . The result follows bearing in mind that , and that for i and m fixed, there are vertices of type , and vertices of type , . We are done. □
Theorem 7. For , .
Proof. Since , and the number of loops in the quiver equals then . We are done. □
The sequence appears encoded in the OEIS as A100104.