# symExt -- from linear presentation matrices over S to linear presentation matrices over E and conversely

## Synopsis

• Usage:
symExt(m,E)
• Inputs:
• m, , a linear presentation matrix over S
• E, a ring, the Koszul dual ring of S
• Outputs:
• , the corresponding linear presentation matrix over E

## Description

Same method as in the single factor case

 i1 : n={1,2} o1 = {1, 2} o1 : List i2 : (S,E) = productOfProjectiveSpaces n o2 = (S, E) o2 : Sequence i3 : vars S, vars E o3 = (| x_(0,0) x_(0,1) x_(1,0) x_(1,1) x_(1,2) |, | e_(0,0) e_(0,1) e_(1,0) ------------------------------------------------------------------------ e_(1,1) e_(1,2) |) o3 : Sequence i4 : m=map(S^4,S^{{ -1,0},{0,-1}}, transpose matrix{{S_0,S_1,0,0},{S_2,0,S_3,S_4}}) o4 = | x_(0,0) x_(1,0) | | x_(0,1) 0 | | 0 x_(1,1) | | 0 x_(1,2) | 4 2 o4 : Matrix S <--- S i5 : mE=symExt(m,E) o5 = {-1, 0} | 0 e_(0,0) 0 0 | {-1, 0} | 0 0 e_(0,0) 0 | {-1, 0} | 0 0 0 e_(0,0) | {-1, 0} | e_(0,1) 0 0 0 | {-1, 0} | -e_(0,0) e_(0,1) 0 0 | {-1, 0} | 0 0 e_(0,1) 0 | {-1, 0} | 0 0 0 e_(0,1) | {0, -1} | 0 e_(1,0) 0 0 | {0, -1} | 0 0 e_(1,0) 0 | {0, -1} | 0 0 0 e_(1,0) | {0, -1} | e_(1,1) 0 0 0 | {0, -1} | 0 e_(1,1) 0 0 | {0, -1} | -e_(1,0) 0 e_(1,1) 0 | {0, -1} | 0 0 0 e_(1,1) | {0, -1} | e_(1,2) 0 0 0 | {0, -1} | 0 e_(1,2) 0 0 | {0, -1} | 0 0 e_(1,2) 0 | {0, -1} | -e_(1,0) 0 0 e_(1,2) | 18 4 o5 : Matrix E <--- E

## Ways to use symExt :

• "symExt(Matrix,Ring)"

## For the programmer

The object symExt is .