🍟 Wargaming with Quadratic Forms and Brauer Configuration Algebras

Recently, Postnikov introduced Bert Kostant’s game to build the maximal positive root associated with the quadratic form of a simple graph. This result, and some other games based on Cartan matrices, give a new version of Gabriel’s theorem regarding algebras classification. In this paper, as a variation of Bert Kostant’s game, we introduce a wargame based on a missile defense system (MDS). In this case, missile trajectories are interpreted as suitable paths of a quiver (directed graph). The MDS protects a region of the Euclidean plane by firing missiles from a ground-based interceptor (GBI) located at the point (0,0). In this case, a missile success interception occurs if a suitable positive number associated with the launches of the enemy army can be written as a mixed sum of triangular and square numbers.

A vector such that is called a root of .
The quadratic form of a quiver has the form: . If , then the Euler quadratic form of , where, . These quadratic forms coincide if is acyclic and connected.
For instance, the quadratic form of is given by the equality . Note that, are positive roots of .
The reflection at a vertex of a finite, connected, and acyclic quiver is given by In terms of the coordinates of , we see that has coordinates if , .
The Weyl group is the group of automorphisms of generated by the set of reflections .
Excited if .
Goal: Make everyone happy or excited.
The game is played as follows: Initially no chips are present (hence for all and all vertices are happy). Then we place a chip at vertex , so is exci te d but n eig hbo rs of are unhappy.
Subsequently, do the following ''reflection'': Pick an unhappy vertex , and replace by .
Some Properties of BCA's (Green and Schroll , 2017) There is a bijective correspondence between the set of indecomposable projective modules over and polygons in .
The BCA is a multiserial algebra.
The number of summands in the heart of an indecomposable projective module over with radical square distinct of zero equals the number of non-truncated vertices of the polygons corresponding to counting repetitions.
If is an indecomposable projective module over corresponding to a polygon then the radical of is a sum of uniserial modules, where is the number of nontruncated vertices of and where the intersection of any two of the uniserial modules is a simple module.
Let be a Brauer configuration algebra associated to the Brauer configuration and let be a full set of equivalence class representatives of special cycles.
Assume that for is a special -cycle where is a non-truncated vertex in then where denotes the number of vertices of , denotes the number of arrows in the -cycle and .

On the center of a Brauer Configuration Algebra
Let be a reduced (i.e, without truncated vertices) and connected Brauer configuration and let be its induced quiver and let be the induced Brauer configuration algebra such that , then the dimension of the center of denoted is given by the formula (Sierra, 2017) where .
Wargaming with admissible paths.
Definition. If then they are equivalent.
Thus, subsets constitute a partition of . , i.e. polygons are representative of classes admissible paths, whose associated word is given by the corresponding slope sequence.
for any .

If
, where denotes a representative of a class of admissible paths. Thus, an ordering is defined in such a way that in successor sequences, it holds that .
The game we define is similar to the way of a missile defense system (MDS) works.