Topics in Fomin-Kirillov Algebra

We introduce the notion of being 'z-star' for homogeneous polynomials. By proving a theorem plus developing a conjecture we state that, with a graded lexicographic monomial ordering, the reduced Grobner basis, for the ideal I generated by the relations of Fomin-Kirilov algebra FK(n), consists of 'z-star' polynomials. For general n, we find the character of the Fomin-Kirillov a lgebra, for some finite usual degrees with general set partition degree. We find the decomposition of this character in irreducible characters where this decomposition stabilizes at some enough big n. We develop a quotient of FK(n), denoted by FKCn (n), by making the quotient of the free algebra generated by the edges of an n-cycle, compared to the associated complete graph, where the ideal is generated by the relations of FK(n) except for letting the missing edges equal to zero and keeping only the edges of the polygon. We find the character map of algebra FKCn (n) and prove that the dimension of it equals the Lucas Number Ln and its Hilbert series is q-Lucas polynomial. We consider the commutative quotient of Fomin-Kirilov algebra, denoted by FK(n) and find its Grobner basis.