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<head><title>Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space</title>
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<div><h1>Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>Grassmannian(k,r)</tt><br/><tt>Grassmannian(k,r,R)</tt></div>
</dd></dl>
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</li>
<li><span>Function: <a href="___Grassmannian_lp__Z__Z_cm__Z__Z_rp.html" title="the Grassmannian of linear subspaces of a vector space">Grassmannian</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>k</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>r</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, the ideal of the Grassmannian variety of all projective <tt>k</tt>-planes in <b>P</b><sup>r</sup></span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>CoefficientRing => </tt><span><span>a <a href="___Ring.html">ring</a></span>, <span>default value ZZ</span>, the coefficient ring for the polynomial ring to be made</span></span></li>
<li><span><tt>Variable => </tt><span><span>a <a href="___Symbol.html">symbol</a></span>, <span>default value p</span>, the base symbol for the indexed variables to be used. The subscripts are the elements of <tt>subsets(n+1,k+1)</tt>, converted to sequences and, if <tt>k</tt> is 0, converted to integers.</span></span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div>If a polynomial ring <tt>R</tt> is given as the third argument, then the resulting ideal is moved to that ring.<table class="examples"><tr><td><pre>i1 : Grassmannian(1,3)
o1 = ideal(p p - p p + p p )
1,2 0,3 0,2 1,3 0,1 2,3
o1 : Ideal of ZZ[p , p , p , p , p , p ]
0,1 0,2 1,2 0,3 1,3 2,3</pre>
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<tr><td><pre>i2 : J = Grassmannian(2,5, CoefficientRing => ZZ/31, Variable => T)
o2 = ideal (T T - T T + T T , T T -
2,3,5 1,4,5 1,3,5 2,4,5 1,2,5 3,4,5 2,3,4 1,4,5
------------------------------------------------------------------------
T T + T T , T T - T T + T T ,
1,3,4 2,4,5 1,2,4 3,4,5 2,3,5 0,4,5 0,3,5 2,4,5 0,2,5 3,4,5
------------------------------------------------------------------------
T T - T T + T T , T T - T T
1,3,5 0,4,5 0,3,5 1,4,5 0,1,5 3,4,5 1,2,5 0,4,5 0,2,5 1,4,5
------------------------------------------------------------------------
+ T T , T T - T T + T T , T T
0,1,5 2,4,5 2,3,4 0,4,5 0,3,4 2,4,5 0,2,4 3,4,5 1,3,4 0,4,5
------------------------------------------------------------------------
- T T + T T , T T - T T +
0,3,4 1,4,5 0,1,4 3,4,5 1,2,4 0,4,5 0,2,4 1,4,5
------------------------------------------------------------------------
T T , T T - T T + T T - T T ,
0,1,4 2,4,5 1,2,3 0,4,5 0,2,3 1,4,5 0,1,3 2,4,5 0,1,2 3,4,5
------------------------------------------------------------------------
T T - T T + T T , T T - T T
2,3,4 1,3,5 1,3,4 2,3,5 1,2,3 3,4,5 1,2,5 0,3,5 0,2,5 1,3,5
------------------------------------------------------------------------
+ T T , T T - T T + T T , T T
0,1,5 2,3,5 2,3,4 0,3,5 0,3,4 2,3,5 0,2,3 3,4,5 1,3,4 0,3,5
------------------------------------------------------------------------
- T T + T T , T T - T T +
0,3,4 1,3,5 0,1,3 3,4,5 1,2,4 0,3,5 0,2,4 1,3,5
------------------------------------------------------------------------
T T + T T , T T - T T + T T ,
0,1,4 2,3,5 0,1,2 3,4,5 1,2,3 0,3,5 0,2,3 1,3,5 0,1,3 2,3,5
------------------------------------------------------------------------
T T - T T + T T , T T - T T
2,3,4 1,2,5 1,2,4 2,3,5 1,2,3 2,4,5 1,3,4 1,2,5 1,2,4 1,3,5
------------------------------------------------------------------------
+ T T , T T - T T + T T +
1,2,3 1,4,5 0,3,4 1,2,5 0,2,4 1,3,5 0,1,4 2,3,5
------------------------------------------------------------------------
T T - T T + T T , T T - T T
0,2,3 1,4,5 0,1,3 2,4,5 0,1,2 3,4,5 2,3,4 0,2,5 0,2,4 2,3,5
------------------------------------------------------------------------
+ T T , T T - T T + T T +
0,2,3 2,4,5 1,3,4 0,2,5 0,2,4 1,3,5 0,2,3 1,4,5
------------------------------------------------------------------------
T T , T T - T T + T T , T T -
0,1,2 3,4,5 0,3,4 0,2,5 0,2,4 0,3,5 0,2,3 0,4,5 1,2,4 0,2,5
------------------------------------------------------------------------
T T + T T , T T - T T + T T ,
0,2,4 1,2,5 0,1,2 2,4,5 1,2,3 0,2,5 0,2,3 1,2,5 0,1,2 2,3,5
------------------------------------------------------------------------
T T - T T + T T - T T , T T
2,3,4 0,1,5 0,1,4 2,3,5 0,1,3 2,4,5 0,1,2 3,4,5 1,3,4 0,1,5
------------------------------------------------------------------------
- T T + T T , T T - T T +
0,1,4 1,3,5 0,1,3 1,4,5 0,3,4 0,1,5 0,1,4 0,3,5
------------------------------------------------------------------------
T T , T T - T T + T T , T T -
0,1,3 0,4,5 1,2,4 0,1,5 0,1,4 1,2,5 0,1,2 1,4,5 0,2,4 0,1,5
------------------------------------------------------------------------
T T + T T , T T - T T + T T ,
0,1,4 0,2,5 0,1,2 0,4,5 1,2,3 0,1,5 0,1,3 1,2,5 0,1,2 1,3,5
------------------------------------------------------------------------
T T - T T + T T , T T - T T
0,2,3 0,1,5 0,1,3 0,2,5 0,1,2 0,3,5 1,2,4 0,3,4 0,2,4 1,3,4
------------------------------------------------------------------------
+ T T , T T - T T + T T , T T
0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4
------------------------------------------------------------------------
- T T + T T , T T - T T +
0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4
------------------------------------------------------------------------
T T , T T - T T + T T )
0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4
ZZ
o2 : Ideal of --[T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T ]
31 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 0,1,5 0,2,5 1,2,5 0,3,5 1,3,5 2,3,5 0,4,5 1,4,5 2,4,5 3,4,5</pre>
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The variables of the ring are based on the symbol provided, but assignments are not made until the ring or the ideal is assigned to a global variable or is submitted to <a href="_use.html" title="install or activate object">use</a>, as follows.<table class="examples"><tr><td><pre>i3 : T_(0,2,3)
o3 = T
0,2,3
o3 : IndexedVariable</pre>
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<tr><td><pre>i4 : use ring J
ZZ
o4 = --[T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T ]
31 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 0,1,5 0,2,5 1,2,5 0,3,5 1,3,5 2,3,5 0,4,5 1,4,5 2,4,5 3,4,5
o4 : PolynomialRing</pre>
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<tr><td><pre>i5 : T_(0,2,3)
o5 = T
0,2,3
ZZ
o5 : --[T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T , T ]
31 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 0,1,5 0,2,5 1,2,5 0,3,5 1,3,5 2,3,5 0,4,5 1,4,5 2,4,5 3,4,5</pre>
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In many ways, more natural than returning an ideal would be to return the corresponding quotient ring or variety, but creating a quotient ring involves computing a Gröbner basis, which might impose a heavy computational burden that the user would prefer to avoid.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Schubert_lp__Z__Z_cm__Z__Z_cm__Visible__List_rp.html" title="find the Pluecker ideal of a Schubert variety">Schubert</a> -- find the Pluecker ideal of a Schubert variety</span></li>
<li><span><a href="_pfaffians.html" title="ideal generated by Pfaffians">pfaffians</a> -- ideal generated by Pfaffians</span></li>
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