Rectagraphs from root systems

One can construct (0,2)-graphs Γ(u) from root systems with simply laced diagram. This may or may not be relevant to computation of mod p cohomology.

The structure here is richer: not only a (0,2)-graph, but also a grading.

Cohomology of root systems

Let Φ be a root system, let k be any field, and let C_i = ⋀ⁱ Φ⁺ be the i-th exterior power of the k-vectorspace with basis Φ⁺. Consider the cochain complex ... ← C_i+1 ← C_i ← ... with coboundary operator d defined by

d_i(a₁ ∧ ... ∧ a_i) = Σ_j Σ_{a_j=b+c} (–1)^j f(b,c) a₁ ∧ ... ∧ a_j–1 ∧ b ∧ c ∧ a_j+1 ∧ ... ∧ a_i,

where f(b,c)=–f(c,b) and the sum is over unordered pairs of positive roots {b,c} with b+c = a_j. One checks that indeed d_i+1d_i = 0 provided that f(b,c)f(b+c,d) = f(b,c+d)f(c,d). (For a simply laced root system one can find such an f that only takes the values 1 or –1, e.g. by borrowing it from the corresponding Lie algebra: [e_r,e_s] = f(r,s)e_r+s.)

This complex is the direct sum of such complexes where the basis vectors are restricted to (the exterior products of the elements of) the vertices of Γ(u). For each u we can ask for the cohomology of this complex.

In characteristic 0 the only contribution to cohomology is provided by the Γ(u) that have a single vertex only. In characteristic p there may be further cohomology.

For example, H^p has further mod p cohomology in A_p+1. Look at the sum vector u = (1,2,3,...,p,1). The corresponding graph has 2^p vertices and valency p, that is, is the p-cube. The matrix for d_p has rank p but p-rank p-1, so that H^p has larger dimension in characteristic p than in characteristic 0.

In characteristic 0, the Poincaré polynomial P(t) = Σ dim H_i tⁱ equals ∏_d ((t^d-1)/(t-1)) where d runs over the degrees: that is the sequence 2,3,...,n+1 for A_n and 2,4,6,...,2n-4,2n-2,n for D_n. This explains the p=0 data below.

Data: A_n, B_n, C_n, D_n, F₄. G₂. See also On Kostant's theorem for Lie algebra cohomology, by the University of Georgia VIGRE algebra group (preprint, 2007).

A_n

A₀:

p		H⁰
0		1

A₁:

p H⁰ H¹

0 1 1

A₂:

p H⁰ H¹ H² H³

0 1 2 2 1

A₃:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶

0 1 3 5 6 5 3 1

2 1 3 6 8 6 3 1

A₄:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ H¹⁰

0 1 4 9 15 20 22 20 15 9 4 1

2 1 4 11 25 38 42 38 25 11 4 1

3 1 4 9 17 25 28 25 17 9 4 1

A₅:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷, H⁸ ... H¹⁵

0 1 5 14 29 49 71 90 101 ... 1

2 1 5 17 52 119 209 308 381 ... 1

3 1 5 14 33 66 110 151 172 ... 1

A₆:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ H¹⁰, H¹¹ ... H²¹

0 1 6 20 49 98 169 259 359 455 531 573 ... 1

2 1 6 24 88 263 630 1290 2293 3523 4657 5313 ... 1

3 1 6 20 55 131 274 505 802 1114 1396 1576 ... 1

5 1 6 20 49 98 173 280 414 549 650 700 ... 1

A₇:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ H¹⁰ H¹¹ H¹² H¹³ H¹⁴ H¹⁵ ... H²⁸

0 1 7 27 76 174 343 602 961 1415 1940 2493 3017 3450 3736 3836 3736 ... 1

2 1 7 32 134 479 1433 3732 8543 17384 31600 51128 73885 96110 112822 119116 112822 ... 1

3 1 7 27 84 227 556 1249 2490 4392 7045 10452 14302 17856 20252 21064 20252 ... 1

5 1 7 27 76 174 351 657 1153 1869 2773 3783 4790 5676 6306 6538 6306 ... 1

B_n

B₂:

p	H⁰	H¹	H²	H³	H⁴
0	1	2	2	2	1
2	1	3	4	3	1

B₃:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹

0 1 3 5 7 8 8 7 5 3 1

2 1 4 12 24 33 33 24 12 4 1

3 1 3 6 10 12 12 10 6 3 1

B₄:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ ... H¹⁶

0 1 4 9 16 24 32 39 44 46 44 ... 1

2 1 5 22 70 168 336 555 732 794 732 ... 1

3 1 4 10 23 44 74 114 147 158 147 ... 1

5 1 4 9 17 29 44 57 63 64 63 ... 1

B₅:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ H¹⁰ H¹¹ H¹², H¹³ ... H²⁵

0 1 5 14 30 54 86 125 169 215 259 297 325 340 ... 1

2 1 6 30 129 430 1225 3082 6624 12219 19945 28994 37050 41621 ... 1

3 1 5 15 40 94 208 430 790 1320 1993 2639 3136 3441 ... 1

5 1 5 14 31 61 113 197 312 443 574 694 798 869 ... 1

7 1 5 14 30 55 92 144 212 291 367 424 456 469 ... 1

C_n

C₂ = B₂.

C₃:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹

0 1 3 5 7 8 8 7 5 3 1

2 1 5 13 24 33 33 24 13 5 1

3 1 3 6 10 12 12 10 6 3 1

C₄:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ ... H¹⁶

0 1 4 9 16 24 32 39 44 46 44 ... 1

2 1 7 27 79 187 363 581 767 840 767 ... 1

3 1 4 10 22 43 79 124 151 156 151 ... 1

5 1 4 9 16 26 41 57 67 70 67 ... 1

C₅:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ H¹⁰ H¹¹ H¹², H¹³ ... H²⁵

0 1 5 14 30 54 86 125 169 215 259 297 325 340 ... 1

2 1 9 45 174 562 1544 3691 7728 14158 22776 32374 40853 45873 ... 1

3 1 5 15 39 93 218 466 847 1372 2034 2708 3277 3645 ... 1

5 1 5 14 30 59 114 203 319 448 581 704 790 828 ... 1

7 1 5 14 30 54 86 128 188 272 367 443 481 491 ... 1

D_n

D₃ = A₃.

D₄:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ H¹⁰ H¹¹ H¹²

0 1 4 9 16 23 28 30 28 23 16 9 4 1

2 1 4 15 37 67 103 122 103 67 37 15 4 1

3 1 4 9 17 28 39 44 39 28 17 9 4 1

D₅:

p H⁰ H¹ H² H³ H⁴ H⁵ H⁶ H⁷ H⁸ H⁹ H¹⁰ H¹¹ ... H²⁰

0 1 5 14 30 54 85 120 155 185 205 212 205 ... 1

2 1 5 22 79 216 516 1072 1815 2586 3247 3530 3247 ... 1

3 1 5 14 37 85 163 285 449 616 746 798 746 ... 1

5 1 5 14 30 54 89 143 213 273 303 310 303 ... 1

F₄

F₄:

p	H⁰	H¹	H²	H³	H⁴	H⁵	H⁶	H⁷	H⁸	H⁹	H¹⁰	H¹¹	H¹²	H¹³	...	H²⁴
0	1	4	9	16	25	36	48	60	71	80	87	92	94	92	...	1
2	1	6	30	118	371	1008	2381	4791	8379	13074	18005	21600	22860	21600	...	1
3	1	4	12	37	90	189	369	654	1051	1526	1980	2311	2436	2311	...	1
5	1	4	9	17	31	53	82	127	197	277	340	368	372	368	...	1
7	1	4	9	16	25	38	60	94	133	163	173	165	158	165	...	1

G₂

G₂:

p	H⁰	H¹	H²	H³	H⁴	H⁵	H⁶
0	1	2	2	2	2	2	1
2	1	3	6	8	6	3	1
3	1	3	6	8	6	3	1

Rectagraphs from root systems

Cohomology of root systems

An

Bn

Cn

Dn

F4

G2

A_n

B_n

C_n

D_n

F₄

G₂