# 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 Ci = ⋀i Φ+ be the i-th exterior power of the k-vectorspace with basis Φ+. Consider the cochain complex ... ← Ci+1 ← Ci ← ... with coboundary operator d defined by
di(a1 ∧ ... ∧ ai) = Σj Σaj=b+c (–1)j f(b,c) a1 ∧ ... ∧ aj–1bcaj+1 ∧ ... ∧ ai,
where f(b,c)=–f(c,b) and the sum is over unordered pairs of positive roots {b,c} with b+c = aj. One checks that indeed di+1di = 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: [er,es] = f(r,s)er+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, Hp has further mod p cohomology in Ap+1. Look at the sum vector u = (1,2,3,...,p,1). The corresponding graph has 2p vertices and valency p, that is, is the p-cube. The matrix for dp has rank p but p-rank p-1, so that Hp has larger dimension in characteristic p than in characteristic 0.

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

Data: An, Bn, Cn, Dn, F4. G2. See also On Kostant's theorem for Lie algebra cohomology, by the University of Georgia VIGRE algebra group (preprint, 2007).

## An

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

## Bn

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

## Cn

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

## Dn

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