OFFSET
0,6
COMMENTS
Cl(p,q)(R) is a 2^(p+q)-dimensional algebraic structure generated by {e_1, e_2, ..., e_(p+q)}, where (e_1)^2 = (e_2)^2 = ... = (e_p)^2 = +1, (e_(p+1))^2 = (e_(p+2))^2 = ... = (e_q)^2 = -1, (e_i)*(e_j) = -(e_j)*(e_i) for any i != j (anti-commutativity), ((e_i)*(e_j))*(e_k) = (e_i)*((e_j)*(e_k)) for any i, j, k (associativity). So the 2^(p+q) basis are all elements of the form Product_{s=1..t} e_(i_s) where 1 <= i_1 < i_2 < ... < i_t <= p + q and {i_1, i_2, ..., i_t} runs through all 2^(p+q) subsets of {1, 2, ..., p + q} (due to the failure of commutativity, one should be careful when taking continued products). Examples include: the real numbers Cl(0,0)(R), the complex numbers Cl(0,1)(R), split-complex numbers Cl(1,0)(R), quaternions Cl(1,0)(R), etc. If p + q = p' + q', then Cl(p,q)(R) is equal to Cl(p',q')(R) if and only if T(p,q) = T(p',q').
It can be shown that (Product_{s=1..t} e_(i_s))^2 = (-1)^(t*(t-1)/2)*(Product_{s=1..t} (e_(i_s))^2). So (Product_{s=1..t} e_(i_s))^2 = -1 if and only if t == 0, 1 (mod 4) and #({i_1, i_2, ..., i_t} intersect {p + 1, p + 2, ..., p + q}) is odd, or t == 2, 3 (mod 4) and #({i_1, i_2, ..., i_t} intersect {p + 1, p + 2, ..., p + q}) is even.
In general, let A = (a_ij) be any n X n symmetric {-1,1}-matrix, we can define an algebraic structure generated by {e_1, e_2, ..., e_n} where (e_i)^2 = a_ii for i = 1..n, (e_i)*(e_j) = (a_ij)*(e_j)*(e_i) for any i != j, ((e_i)*(e_j))*(e_k) = (e_i)*((e_j)*(e_k)) for any i, j, k. Clifford algebras are the cases where a_ij = -1 for any i != j. It can be shown that (Product_{s=1..t} e_(i_s)) * (Product_{s'=1..u} e_(j_s')) = (Product_{s=1..t, s'=1..u, i_s>=j_s'} a_(i_s)(j_s')) * (Product_{s=1..v} e_(k_s)), where 1 <= k_1 < k_2 < ... < k_v <= n and {k_1, k_2, ..., k_v} is the symmetric difference between {i_1, i_2, ..., i_t} and {j_1, j_2, ..., j_u}. Specially, (Product_{s=1..t} e_(i_s)))^2 = Product_{1<=s'<=s<=n} a_ss'. The 2^n basis, together with their additive inverses, form a group of order 2^(n+1) under multiplication, which is abelian if and only if a_ij = 1 for any i != j (in this case, it is isomorphic to (C_2)^(n+1) if a_ii = 1 for i = 1..n, and (C_2)^(n-1) X C_4 otherwise). The structure of this group can be complicated. For example, when n = 2, it can be isomorphic to either (C_2)^3, C_2 X C_4, C_2 X D_4 or Q_8.
LINKS
Jianing Song, Antidiagonals n = 0..99, flattened
Wikipedia, Clifford algebras
FORMULA
T(p,q) = Sum_{i=0..p} Sum_{j=0..q} binomial(p, i)*binomial(q, j)*(binomial(i - j, 2) mod 2).
T(p,q) = 2^(p+q) - A323346(p,q).
EXAMPLE
Table begins
p\q| 0 1 2 3 4 5 ...
---+-------------------------------
0 | 0, 1, 3, 6, 10, 16, ...
1 | 0, 1, 4, 10, 20, 36, ...
2 | 1, 2, 6, 16, 36, 72, ...
3 | 4, 6, 12, 28, 64, 136, ...
4 | 10, 16, 28, 56, 120, 256, ...
5 | 20, 36, 64, 120, 240, 496, ...
...
Example for T(1,3) = 10: (Start)
1^2 = 1;
(e_1)^2 = 1;
(e_2)^2 = -1;
(e_3)^2 = -1;
(e_4)^2 = -1;
((e_1)*(e_2))^2 = -(e_1)^2*(e_2)^2 = 1;
((e_1)*(e_3))^2 = -(e_1)^2*(e_3)^2 = 1;
((e_1)*(e_4))^2 = -(e_1)^2*(e_4)^2 = 1;
((e_2)*(e_3))^2 = -(e_2)^2*(e_3)^2 = -1;
((e_2)*(e_4))^2 = -(e_2)^2*(e_4)^2 = -1;
((e_3)*(e_4))^2 = -(e_3)^2*(e_4)^2 = -1;
((e_1)*(e_2)*(e_3))^2 = -(e_1)^2*(e_2)^2*(e_3)^2 = -1;
((e_1)*(e_2)*(e_4))^2 = -(e_1)^2*(e_2)^2*(e_4)^2 = -1;
((e_1)*(e_3)*(e_4))^2 = -(e_1)^2*(e_3)^2*(e_4)^2 = -1;
((e_2)*(e_3)*(e_4))^2 = -(e_2)^2*(e_3)^2*(e_4)^2 = 1;
((e_1)*(e_2)*(e_3)*(e_4))^2 = (e_1)^2*(e_2)^2*(e_3)^2*(e_4)^2 = -1. (End)
From Peter Luschny, Jan 13 2019: (Start)
The first few lines of the triangle T(i-j,j) are:
[0] 0;
[1] 0, 1;
[2] 1, 1, 3;
[3] 4, 2, 4, 6;
[4] 10, 6, 6, 10, 10;
[5] 20, 16, 12, 16, 20, 16;
[6] 36, 36, 28, 28, 36, 36, 28;
[7] 64, 72, 64, 56, 64, 72, 64, 56;
[8] 120, 136, 136, 120, 120, 136, 136, 120, 120;
[9] 240, 256, 272, 256, 240, 256, 272, 256, 240, 256; (End)
MAPLE
s := sqrt(2): h := n -> [ 0, -s, -2, -s, 0, s, 2, s][1 + modp(n+1, 8)]:
T := proc(n, k) option remember;
if n = 0 then return 2^(k - 1) + 2^((k - 3)/2)*h(k + 2) fi;
if k = 0 then return 2^(n - 1) + 2^((n - 3)/2)*h(n) fi;
T(n, k-1 ) + T(n-1, k) end:
for n from 0 to 9 do seq(T(n, k), k=0..9) od; # Peter Luschny, Jan 12 2019
MATHEMATICA
T[n_, k_] := Sum[Binomial[n, i] Binomial[k, j] Mod[Binomial[i - j, 2], 2], {i, 0, n}, {j, 0, k}];
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 19 2019 *)
PROG
(PARI) T(p, q) = sum(i=0, p, sum(j=0, q, binomial(p, i)*binomial(q, j)*(binomial(i-j, 2)%2)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Jianing Song, Jan 04 2019
STATUS
approved