Symmetric monomial ideals

In recent years, the study of chains of ideals of increasingly larger polynomial rings gain considerable attention. These ideals possess certain symmetries arise naturally in various areas of mathematics, including algebraic chemistry, algebraic statistics, toric algebra and group theory. A newly discovered technique to study such a chain is to pass to a non-Noetherian limit of the chain. Typically, this leads to the study of ideals in a polynomial ring with infinite many variables. Recently, significant advances have been made in this new research direction. Aschenbrenner-Hillar [1] introduced a completely new algebraic structure Sym( )-Noetherian. It provides a framework as well as a motivation for further studies of properties of Sym(\infty )-invariant ideals.

Although is not a Noetherian ring, a number of useful finiteness results have been established for this ring. For instance, it is known that Sym-invariant ideals in satisfy the ascending chain condition, or in other words, R is Sym-Noetherian. This celebrated result was first discovered by Cohen in his investigation of the variety of metabelian groups. Based on the Sym-Noetherianity of , Nagel and Römer introduced Hilbert series for Sym-invariant chains and showed that they are rational functions.

As a consequence, they determined the asymptotic behaviors of the codimension and multiplicity along Sym-invariant chains: the codimension grows eventually linearly, whereas the multiplicity grows eventually exponentially. This leads to the following general problem:

· Study the asymptotic behavior of invariants along Sym-invariant chains of monomial ideals.

The study of asymptotic behavior of Inc-invariant chains is in its early stage and many interesting problems are still open. We will be considering the following problems.

· Characterize those - or -invariant chains of square-free monomial ideals for which is Cohen-Macaulay whenever n≫0.

· Let the chain be Sym- or Inc-invariant. Study the primary decomposition of .

· Study the asymptotic behavior of Betti tables of square-free monomial ideals of Sym- or Inc-invariant chains.

The proposed project is about exploring the recently developed theories and gets the attention of top researchers across the globe. These researches are gaining places in the top ranked mathematics journals. Much of the commentary on mathematics and science in Pakistan focuses on national economic competitiveness and the economic well-being of citizens and enterprises. There is reason enough for concern about these matters, but it is yet more fundamental to recognize that the need of nurturing culture of research in mathematics in the country. This project serves toward establishing fundamental research in algebra of highly regarded mathematics journals like the one mentioned below.

In a recent survey Le, Kubetzke and Römer recorded all the recent development and enlisted the following conjectures.

1. Characterize those -invariant chains of square-free monomial ideals for which is Cohen-Macaulay whenever .

2. Study the primary decomposition of , is an -invariant chain of square-free monomial ideals.

3. Study the asymptotic behavior of Betti tables of square-free monomial ideals of Sym- or Inc-invariant chains.

4. Let be an -invariant chain of ideals. Then the projective dimension is eventually a linear function, that is, there exist integers and such that whenever .

5. Let be a nonzero -invariant chain of graded ideals. Then is eventually a linear function, that is, there exist integers and such that whenever .

They found affirmative answers of these conjectures for certain classes of monomial ideals, but these conjectures are widely open. From our initial investigation, we come up with an independent approach to study the -invariant chain of square-free monomial ideal through -invariant chain of simplicial complexes. It leads to the following natural questions:

6. Study the asymptotic behavior of -invariant chain of square-free monomial ideals in context of -invariant chain of facet complexes.

7. Study the asymptotic behavior of primary decomposition of -invariant chain of square-free monomial ideals in context of -invariant chain of face complexes.

8. Study the relation between the f-vector and h-vector of and for -invariant chain of square-free monomial ideals.

The potential aim of this project is to investigate the asymptotic behavior of -invariant chain of square-free monomial ideals associated to simplicial complexes. This research will extend the study of Stanley-Reisner ring and Facet ideal rings in symmetric Noetherian structure. We are hopeful that we will be successful in proving the above stated conjectures but also establishing a new combinatorial study of -invariant chain of square-free monomial ideal through -invariant chain of simplicial complexes.