# flattenedGenericTensor -- Make a generic tensor of given format

## Synopsis

• Usage:
flattenedGenericTensor(L,kk)
• Inputs:
• L, a list, of positive ZZ
• kk, a ring, Name of ground field (or ring)
• Outputs:

## Description

Given a list $L = \{a, b_1,\dots, b_n\}$ of positive integers with $a= sum_i b_i,$ and a field (or ring of integers) kk, the script creates a polynomial ring $S$ over $kk$ with $a\times b_1\times\cdots\times b_n$ variables, and a generic map $$f: A \to B_1\otimes\cdots \otimes B_n$$ of LabeledModules over $S$, where $A$ is a free LabeledModule of rank $a$ and $B_i$ is a free LabeledModule of rank $b_i$. We think of $f$ as representing a tensor of type $(a,b_1,\dots,b_n)$ made from the elementary symmetric functions.

The format of $F$ is the one required by tensorComplex1, namely $f: A \to B_1\otimes \cdots \otimes B_n$, with $a = rank A, b_i = rank B_i$.

 i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing i2 : f = flattenedGenericTensor({5,2,1,2},kk) o2 = | x_(0,0,0,0) x_(1,0,0,0) x_(2,0,0,0) x_(3,0,0,0) x_(4,0,0,0) | | x_(0,0,0,1) x_(1,0,0,1) x_(2,0,0,1) x_(3,0,0,1) x_(4,0,0,1) | | x_(0,1,0,0) x_(1,1,0,0) x_(2,1,0,0) x_(3,1,0,0) x_(4,1,0,0) | | x_(0,1,0,1) x_(1,1,0,1) x_(2,1,0,1) x_(3,1,0,1) x_(4,1,0,1) | 4 5 o2 : Matrix (kk[x ..x ]) <--- (kk[x ..x ]) 0,0,0,0 4,1,0,1 0,0,0,0 4,1,0,1 i3 : numgens ring f o3 = 20 i4 : betti matrix f 0 1 o4 = total: 4 5 -1: . 5 0: 4 . o4 : BettiTally i5 : S = ring f o5 = S o5 : PolynomialRing i6 : tensorComplex1 f o6 = | -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0) 0 -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0) 0 -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0) 0 -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0) 0 -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0) 0 -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0) 0 -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0) 0 -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0) 0 -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0) 0 -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0) 0 | | -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0) -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0) -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0) -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0) -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0) -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0) -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0) -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0) -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0) -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0) | | -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1) -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1) -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1) -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1) -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1) -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1) -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1) -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1) -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1) -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1) -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) | | 0 -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1) 0 -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1) 0 -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1) 0 -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1) 0 -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1) 0 -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1) 0 -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1) 0 -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1) 0 -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1) 0 -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1) | 4 20 o6 : Matrix S <--- S