# Grass -- coordinate ring of a Grassmannian

## Synopsis

• Usage:
Grass(k,n)
Grass(k,n,K)
• Inputs:
• k, an integer
• n, an integer
• K, a ring, optional with default value QQ, the coefficient ring to be used
• Optional inputs:
• Variable => ..., default value "p", specify a name for a variable
• Outputs:
• , the coordinate ring of the Grassmannian variety of all projective $k$-planes in $\mathbb{P}^n$

## Description

This method calls the method Grassmannian, and Grass(k,n,K,Variable=>p) can be considered equivalent to quotient Grassmannian(k,n,Variable=>p,CoefficientRing=>K). However, the method Grass creates no more than an instance of ring for a given tuple (k,n,K,p).

 i1 : R = Grass(2,4,ZZ/11) o1 = R o1 : QuotientRing i2 : R === Grass(2,4,ZZ/11) o2 = true

In order to facilitate comparisons, the outputs of the methods chowForm, hurwitzForm, tangentialChowForm, chowEquations, and dualize always lie in these rings.

 i3 : L = trim ideal(random(1,Grass(0,3,ZZ/11,Variable=>x)),random(1,Grass(0,3,ZZ/11,Variable=>x))) o3 = ideal (x + 2x , x - x + 2x ) 1 3 0 2 3 ZZ o3 : Ideal of --[x ..x ] 11 0 3 i4 : w = chowForm L o4 = x + x + 2x - 2x - 2x 0,1 1,2 0,3 1,3 2,3 ZZ --[x ..x , x , x , x , x ] 11 0,1 0,2 1,2 0,3 1,3 2,3 o4 : -------------------------------------- x x - x x + x x 1,2 0,3 0,2 1,3 0,1 2,3 i5 : ring w === Grass(1,3,ZZ/11,Variable=>x) o5 = true i6 : L' = chowEquations w o6 = ideal (x + 2x , x - x + 2x ) 1 3 0 2 3 ZZ o6 : Ideal of --[x ..x ] 11 0 3 i7 : ring L' === Grass(0,3,ZZ/11,Variable=>x) o7 = true i8 : L''= chowEquations(w,Variable=>y) o8 = ideal (y + 2y , y - y + 2y ) 1 3 0 2 3 ZZ o8 : Ideal of --[y ..y ] 11 0 3 i9 : ring L'' === Grass(0,3,ZZ/11,Variable=>y) o9 = true

## Ways to use Grass :

• "Grass(ZZ,ZZ)"
• "Grass(ZZ,ZZ,Ring)"

## For the programmer

The object Grass is .