The Hibi ideal of P is a MonomialIdeal built over a ring in 2n variables x0, ..., xn-1, y0, ..., yn-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 xi associated with members of I and the yi 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 , x x x y y y , x x x y y y , x x y y 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 0 1 3 2 4 5 0 1 2 3 4 5 0 2 1 3 4 5
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x x y y y y , x y y y y y , y y y y y y )
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
o2 : MonomialIdeal of QQ[x , x , x , x , x , x , y , y , y , y , y , y ]
0 1 2 3 4 5 0 1 2 3 4 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: . . . .
2: . . . .
3: . . . .
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 ith 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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