User:Tomruen/projective spaces

Projective Geometry PG(n,k) is dimension n, size k. Each is self dual with n×n Configuration matrix.

They only exist for k as a prime power, 2,3,4(=2²),5,7,8,9(=3²),11,13,16(=2⁴)..., skipping 6,10,12,14,15,...

Special cases: (Not actually projective spaces)

• PG(n,1) represents the n-simplex family, with S_n+1 symmetry, aut((n+1)!).
• PG(1,k) represent k+1 points. PG(n,k) has PG(n-1,k) facets, recursively backwards, f-vectors seen as row incidences.

Projective planes: (n=2)

• PG(2,k) is configuration [(k²+k+1)_k+1]
• Removing 1 vertex becomes [k(k+1)_k (k²)_k+1], dual [(k²)_k+1 k(k+1)_k]
  • (7₃) --> (6₂ 4₃) --> (4₃ 6₂) Complete quadrilateral
  • (13₄) --> (12₃ 9₄) --> (9₄ 12₃) Hesse configuration

Points and hyperplanes

Points and hyperplanes for (n,k) = (kⁿ⁺¹-1)/(k-1) = kⁿ + k^n-1 + ... + k + 1.

Points (n,k)

k\n	1	2	3	4	5
1	2	3	4	5	6
2	3	7	15	31	63
3	4	13	40	121	364
4	5	21	85	341	1365
5	6	31	156	781	3906
6	7	43	259	1555	9331
7	8	57	400	2801	19608
8	9	73	585	4681	37449

Automorphisms

Automorphisms for k=a^b, a=prime --> aut=(b!)(kⁿ⁺¹-1)(kⁿ⁺¹-k)(kⁿ⁺¹-k²)...(kⁿ⁺¹-kⁿ)/(k-1)

Automorphisms

k\n	1	fact(pow)	2	3	4	5
1	2	1	6	6	24	120
2	6	1	168	20160	9999360	20158709760
3	24	1	5616	12130560	2.37783E+11	4.20648E+16
4	120	2	120960	1974067200	5.16985E+14	2.16786E+21
5	720	1	372000	29016000000	5.66537E+16	2.76612E+24
7	5040	1	5630688	4.63518E+12	1.87035E+20	3.69827E+29
8	40320	6	98896896	2.07351E+14	2.78294E+22	2.39051E+32

Configuration matrices

P(1,1)={ }		P(2,1)={3}			P(3,1)={3,3}				P(4,1)={3,3,3}					P(5,1)={3,3,3,3}
Aut(2)		Aut(6)			Aut(24)				Aut(120)					Aut(720)
2		3	2		4	3	3		5	4	6	4		6	5	10	10	5
		2	3		2	6	2		2	10	3	3		2	15	4	6	4
					3	3	4		3	3	10	2		3	3	20	3	3
									4	6	4	5		4	6	4	15	2
														5	10	10	5	6
P(1,2)=3{ }		P(2,2)			P(3,2)				P(4,2)					P(5,2)
3		7	3		15	7	7		31	15	35	15		63	31	155	155	31
		3	7		3	35	3		3	155	7	7		3	651	15	35	15
					7	7	15		7	7	155	3		7	7	1395	7	7
									15	35	15	31		15	35	15	651	3
														31	155	155	31	63
P(1,3)=4{ }		P(2,3)			P(3,3)				P(4,3)					P(5,3)
4		13	4		40	13	13		121	40	130	40		364	121	1210	1210	121
		4	13		4	130	4		4	1210	13	13		4	11011	40	130	40
					13	13	40		13	13	1210	4		13	13	33880	13	13
									40	130	40	121		40	130	40	11011	4
														121	1210	1210	121	364
P(1,4)=5{ }		P(2,4)			P(3,4)				P(4,4)					P(5,4)
5		21	5		85	21	21		341	85	357	85		1365	341	5797	5797	341
		5	21		5	357	5		5	5797	21	21		5	93093	85	357	85
					21	21	85		21	21	5797	5		21	21	376805	21	21
									85	357	85	341		85	357	85	93093	5
														341	5797	5797	341	1365
P(1,5)=6{ }		P(2,5)			P(3,5)				P(4,5)					P(5,5)
6		31	6		156	31	31		781	156	806	156		3906	781	20306	20306	781
		6	31		6	806	6		6	20306	31	31		6	508431	156	806	156
					31	31	156		31	31	20306	6		31	31	2558556	31	31
									156	806	156	781		156	806	156	508431	6
														781	20306	20306	781	3906