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
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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
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y y y y y y )
0 1 2 3 4 5
o2 : MonomialIdeal of QQ[x ..x , y ..y ]
0 5 0 5
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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: . . . .
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4: . . . .
5: . 10 12 3
o3 : BettiTally
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i4 : betti res (HP^2)
0 1 2 3 4 5
o4 = total: 1 50 100 66 16 1
0: 1 . . . . .
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11: . 50 100 66 16 1
o4 : BettiTally
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i5 : betti res (HP^3)
0 1 2 3 4 5 6
o5 = total: 1 175 450 425 180 33 2
0: 1 . . . . . .
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17: . 175 450 425 180 33 2
o5 : BettiTally
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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
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i12 : apply(#gk, i -> sum(i..<#gk, j -> binomial(j, i) * gk_j))
o12 = {10, 12, 3}
o12 : List
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