Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. We introduce a Macaulay2 package that allows one to deal with classes of monomial ideals in $E$. More precisely, we implement in Macaulay2 some algorithms in order to easy compute stable, strongly stable and lexsegment ideals in $E$. Moreover, an algorithm to check whether an $(n+1)$-tuple $(1, h_1, \ldots, h_n)$ ($h_1 \le n= \dim_K V$) of non--negative integers is the Hilbert function of a graded $K$--algebra of the form $E/I$, with $I$ graded ideal of $E$, is given (Kruskal--Katona's Theorem). In particular, if $H_{E/I}$ is the Hilbert function of a graded $K$-algebra $E/I$ ($I\subset E$ graded ideal), the package is able to construct the unique lexsegment ideal $I^\lex$ such that $H_{E/I} = H_{E/I^\lex}$.

ExteriorIdeals: a package for computing monomial ideals in an exterior algebra

L. Amata;M. Crupi
2018

Abstract

Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. We introduce a Macaulay2 package that allows one to deal with classes of monomial ideals in $E$. More precisely, we implement in Macaulay2 some algorithms in order to easy compute stable, strongly stable and lexsegment ideals in $E$. Moreover, an algorithm to check whether an $(n+1)$-tuple $(1, h_1, \ldots, h_n)$ ($h_1 \le n= \dim_K V$) of non--negative integers is the Hilbert function of a graded $K$--algebra of the form $E/I$, with $I$ graded ideal of $E$, is given (Kruskal--Katona's Theorem). In particular, if $H_{E/I}$ is the Hilbert function of a graded $K$-algebra $E/I$ ($I\subset E$ graded ideal), the package is able to construct the unique lexsegment ideal $I^\lex$ such that $H_{E/I} = H_{E/I^\lex}$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/3131751
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