page_title: Monomial ideal
section_title: Monomial Ideals and Young Diagrams
hierarchical_section_title: Monomial ideal / Monomial Ideals and Young Diagrams
caption_attribution_description: English: A Young diagram and its connection with its monomial ideal.
context_page_description: In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field.
  A toric ideal is an ideal generated by differences of monomials. An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal.
context_section_description: A monomial ideal can be interpreted as a Young diagram. Suppose , then  can be interpreted in terms of the minimal monomials generators as , where  and . The minimal monomial generators of  can be seen as the inner corners of the Young diagram. The minimal generators would determine where we would draw the staircase diagram.
  The monomials not in  lie inside the staircase, and these monomials form a vector space basis for the quotient ring .
  Consider the following example. 
  Let  be a monomial ideal. Then the set of grid points  corresponds to the minimal monomial generators  in . Then as the figure shows, the pink Young diagram consists of the monomials that are not in . The points in the inner corners of the Young diagram, allow us to identify the minimal monomials  in  as seen in the green boxes. Hence, .
  
   
  In general, to any set of grid points, we can associate a Young diagram, so that the monomial ideal is constructed by determining the inner corners that make up the staircase diagram; likewise, given a monomial ideal, we can make up the Young diagram by looking at the  and representing them as the inner corners of the Young diagram. The coordinates of the inner corners would represent the powers of the minimal monomials in . Thus, monomial ideals can be described by Young diagrams of partitions.
  Moreover, the -action on the set of   such that  as a vector space over  has fixed points corresponding to monomial ideals only, which correspond to partitions of size n, which are identified by Young diagrams with n boxes.
caption: A Young diagram and its connection with its monomial ideal.
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