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I'm having trouble setting up the unit cell for a simple monoclinic system. I have my lattice parameters from the paper.

a=12.278
b=13.882
c=6.641
beta=107.75

According to the manual, R'=x1A1+x2A2+x3A3, where A1,A2,A3 are the lattice vectors and R' is the vector of Cartesian coordinates. I thought this meant that my lattice vectors should then be:

A1= a 0 0
A2= 0 b 0
A3=c*cos(beta) 0 c*sin(beta)

or

A1=12.278 0 0
A2=0 13.882 0
A3=-2.0246 0 6.32486

when I put this in the POSCAR file I get the following:

LATTYP: Found a simple monoclinic cell.
ALAT = 15.0906737377
B/A-ratio = 0.9199059128
C/A-ratio = 1.3228787686
COS(beta) = -0.9662049023

Lattice vectors:

A1 = ( -8.2287920000, 0.0000000000, -12.6497200000)
A2 = ( 0.0000000000, -13.8820000000, 0.0000000000)
A3 = ( 6.2041880000, 0.0000000000, 18.9745800000)
Subroutine PRICEL returns:
Original cell was already a primitive cell.

What is going on? COS(beta) is way off and the ratios are way off and A1, A2, and A3 are unrecognizable to me. What am I doing wrong?

1) your POSCAR is correct
2) the lattice information as written in the symmetry part is obtained by transforming the cell such that the 'standard crystallographic' convention is obeyed (which for monoclinic is: the special axis is b, cos gamma <0, and |a| < |c|; the cell is just transformed by rotation addition and subtraction of the lattice vectors, hence the volumes and Bravais lattice types of both cells are the same)
3) for the remaining run of vasp, the Bravais matrix as read from POSCAR (your input) is used

Just to be sure that I understood the posting of "head admin": The POSCAR is read at the beginning and only modified when the symmetry part is executed. Is the k-point generation affected by the symmetry changes? Or that is included in the "remaining run of VASP"?

My question arises in the view of anisotropic k-point meshes within the Monkhorst Pack scheme: The largest unit cell vector should have less k-point partitions. Suppose a cell where c/a is, say 5, then one should use nk x nk x nk/5 (roughly) MP mesh.

However if the unit cell vectors are changed by the symmetry part and then the k-point generation takes place, well, then the k-point subdivision is wrong.

yes: the k-points are generated according to the read Bravais matrix from POSCAR and the symmetry type determined from that input (i.e. according to the lengths of the vectors of the read Bravais Matrix).