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Posets :: hibiRing

hibiRing -- produces the Hibi ring of a poset

Synopsis

Description

The Hibi ring of P is a monomial algebra generated by the monomials which generate the Hibi ideal (hibiIdeal). That is, the monomials built in 2n variables x0, ..., xn-1, y0, ..., yn-1, where n is the size of the ground set of P. The monomials are in bijection with order ideals in P. Let I be an order ideal of P. Then the associated monomial is the product of the xi associated with members of I and the yi associated with non-members of I.

This method returns the toric quotient algebra isomorphic to the Hibi ring. The ideal is the ideal of Hibi relations. The generators of the PolynomialRing H is built over are of the form tI where I is an order ideal of P.
i1 : hibiRing booleanLattice 2

     QQ[t  , t   , t      , t         , t      , t            ]
         {}   {0}   {0, 1}   {0, 1, 2}   {0, 2}   {0, 1, 2, 3}
o1 = ----------------------------------------------------------
                   t   t          - t      t
                    {0} {0, 1, 2}    {0, 1} {0, 2}

o1 : QuotientRing
The Hibi ring of the n chain is just a polynomial ring in n+1 variables.
i2 : hibiRing chain 4

o2 = QQ[t  , t   , t      , t         , t            ]
         {}   {0}   {0, 1}   {0, 1, 2}   {0, 1, 2, 3}

o2 : PolynomialRing
In some cases, it may be faster to use the FourTiTwo method toricGroebner to generate the Hibi relations. Using the Strategy "4ti2" tells the method to use this approach.
i3 : hibiRing(divisorPoset 6, Strategy => "4ti2")
using temporary file name /tmp/M2-14584-0/0

     QQ[t  , t   , t      , t         , t      , t            ]
         {}   {0}   {0, 1}   {0, 1, 2}   {0, 2}   {0, 1, 2, 3}
o3 = ----------------------------------------------------------
                  - t   t          + t      t
                     {0} {0, 1, 2}    {0, 1} {0, 2}

o3 : QuotientRing

See also

Ways to use hibiRing :