# isHilbertSequence(List,Module) -- whether the given sequence is a Hilbert sequence

## Synopsis

• Function: isHilbertSequence
• Usage:
isHilbertSequence(hs,F)
• Inputs:
• hs, a list, a list of integers
• F, , a free module over an exterior algebra
• Outputs:
• , whether the sequence hs satisfies the generalization of the Kruskal-Katona theorem in the free module F

## Description

Let $F$ a free module with homogeneous basis $\{g_1,g_2,\ldots,g_r\},$ with $deg(g_i)=f_i,\ i=1, \ldots, r.$ If $M$ is a graded submodule of F, and $H_{F/M}(t) =\sum_{i=f_1}^{f_r+n}H_{F/M}(i)t^i$ is the Hilbert series of $F/M,$ then the sequence $(H_{F/M}(f_1), H_{F/M}(f_1+1), \ldots, H_{F/M}(f_r+n))\in \mathbb{N}_0^{f_r+n-f_1+1}$ is called the Hilbert sequence of $F/M$ and we denote it by $Hs_{F/M}.$ The integers $f_1, f_1+1, \ldots, f_r+n$ are called the $Hs_{F/M}$-degrees.

Example:

 i1 : E=QQ[e_1..e_4,SkewCommutative=>true] o1 = E o1 : PolynomialRing, 4 skew commutative variables i2 : F=E^{0,0} 2 o2 = E o2 : E-module, free i3 : isHilbertSequence({2,8,3,1,0},F) o3 = true i4 : isHilbertSequence({2,8,3,2,0},F) o4 = false