Recall that a P-partition for a naturally labeled poset P on vertices 1, ..., n is a function f: P →NN which is order-reversing, i.e., if i < j in P then f(i) ≥f(j) in NN. To a P-partition f we can assign the monomial t1f(1) ...tnf(n). The P-partition ring is the ring spanned by the monomials corresponding to P-partitions.
The P-partition ring is more simply generated by the monomials corresponding to the connected order ideals of P. This method returns the toric quotient algebra, whose toric ideal is minimally generated, isomorphic to the P-partition ring.
i1 : P = poset {{1,2}, {2,4}, {3,4}, {3,5}}; |
i2 : pPartitionRing P QQ[t , t , t , t , t , t ] {3} {3, 4} {0} {0, 1} {0, 1, 2, 3} {0, 1, 2, 3, 4} o2 = ----------------------------------------------------------------- t t - t t {3, 4} {0, 1, 2, 3} {3} {0, 1, 2, 3, 4} o2 : QuotientRing |
In some cases, it may be faster to use the FourTiTwo method toricGroebner to generate the toric relations. Using the Strategy "4ti2" tells the method to use this approach.
i3 : pPartitionRing(divisorPoset 6, Strategy => "4ti2") using temporary file name /tmp/M2-27674-0/0 QQ[t , t , t , t , t ] {0} {0, 1} {0, 2} {0, 1, 2} {0, 1, 2, 3} o3 = ----------------------------------------------------- t t - t t {0, 1} {0, 2} {0} {0, 1, 2} o3 : QuotientRing |