# Example: Hibi ideals

The Hibi ideal of $P$ is a MonomialIdeal built over a ring in $2n$ variables $x_0, \ldots, x_{n-1}, y_0, \ldots, y_{n-1}$, where $n$ is the size of the ground set of $P$. The generators of the ideal 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 $x_i$ associated with members of $I$ and the $y_i$ associated with non-members of $I$.

 i1 : P = divisorPoset 12; i2 : HP = hibiIdeal P o2 = monomialIdeal (x x x x x x , x x x x x y , x x x x y y , x x x x y y , 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 4 3 5 0 1 2 3 4 5 ------------------------------------------------------------------------ x x x y y y , x x x y y y , x x y y y y , x x y y y y , x y y y y y , 0 1 3 2 4 5 0 1 2 3 4 5 0 2 1 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 ------------------------------------------------------------------------ y y y y y y ) 0 1 2 3 4 5 o2 : MonomialIdeal of QQ[x ..x , y ..y ] 0 5 0 5

Herzog and Hibi proved that every power of a Hibi ideal has a linear resolution.

 i3 : betti res HP 0 1 2 3 o3 = total: 1 10 12 3 0: 1 . . . 1: . . . . 2: . . . . 3: . . . . 4: . . . . 5: . 10 12 3 o3 : BettiTally i4 : betti res (HP^2) 0 1 2 3 4 5 o4 = total: 1 50 100 66 16 1 0: 1 . . . . . 1: . . . . . . 2: . . . . . . 3: . . . . . . 4: . . . . . . 5: . . . . . . 6: . . . . . . 7: . . . . . . 8: . . . . . . 9: . . . . . . 10: . . . . . . 11: . 50 100 66 16 1 o4 : BettiTally i5 : betti res (HP^3) 0 1 2 3 4 5 6 o5 = total: 1 175 450 425 180 33 2 0: 1 . . . . . . 1: . . . . . . . 2: . . . . . . . 3: . . . . . . . 4: . . . . . . . 5: . . . . . . . 6: . . . . . . . 7: . . . . . . . 8: . . . . . . . 9: . . . . . . . 10: . . . . . . . 11: . . . . . . . 12: . . . . . . . 13: . . . . . . . 14: . . . . . . . 15: . . . . . . . 16: . . . . . . . 17: . 175 450 425 180 33 2 o5 : BettiTally

Moreover, they proved that the projective dimension of the Hibi ideal is the Dilworth number of the poset, i.e., the maximum length of an antichain of $P$.

 i6 : pdim module HP o6 = 2 i7 : dilworthNumber P o7 = 2

They further proved that the $i^{\rm th}$ Betti number of the quotient of a Hibi ideal is the number of intervals of the distributiveLattice of $P$ isomorphic to the rank $i$ boolean lattice. Using an exercise in Stanley's Enumerative Combinatorics'', we recover this instead by looking at the number of elements of the distributive lattice that cover exactly $i$ elements.

 i8 : LP = distributiveLattice P; i9 : cvrs = partition(last, coveringRelations LP); i10 : iCvrs = tally apply(keys cvrs, i -> #cvrs#i); i11 : gk = prepend(1, apply(sort keys iCvrs, k -> iCvrs#k)) o11 = {1, 6, 3} o11 : List i12 : apply(#gk, i -> sum(i..<#gk, j -> binomial(j, i) * gk_j)) o12 = {10, 12, 3} o12 : List