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A215905
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Triangle read by rows in which row n contains the possible dimensions of unital *-subalgebras of the n X n complex matrices.
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4
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1, 1, 2, 4, 1, 2, 3, 5, 9, 1, 2, 3, 4, 5, 6, 8, 10, 16, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 17, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 18, 20, 26, 36, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 25, 26, 27, 29, 37, 49
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OFFSET
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1,3
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COMMENTS
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A *-subalgebra is a subalgebra (i.e., a subspace closed under matrix multiplication) that is closed under the conjugate transpose operation. A unital subalgebra is one that contains the identity matrix.
Every *-subalgebra of the n X n complex matrices is unitarily similar to a direct sum of full matrix algebras of smaller dimension, where elements of those full matrix algebras of the same dimension may or may not be forced to be identical.
As a result of the previous characterization, the possible dimensions of unital *-subalgebras of the n X n matrices are obtained by finding all partitions of n and summing the squares of the parts, possibly omitting some or all duplicated parts.
The first n entries of row n are 1, 2, ..., n. The smallest positive integer not contained in the n-th row is A215914(n).
The last entry of row n is A000290(n).
The 2nd-to-last entry of row n is A002522(n-1).
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LINKS
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EXAMPLE
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When n = 3, there are five different (up to unitary similarity) unital *-subalgebras, which contain matrices of the following forms:
a 0 0 ... a 0 0 ... a 0 0 ... a b 0 ... a b c
0 a 0 ... 0 b 0 ... 0 b 0 ... c d 0 ... d e f
0 0 a ... 0 0 b ... 0 0 c ... 0 0 e ... g h i
The above *-subalgebras have dimensions 1, 2, 3, 5, and 9, which is the 3rd row of the triangle.
When n = 4, we can compute the possible dimension by finding all partitions of 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. Then the possible dimensions are 4^2 = 16, 3^2 + 1^2 = 10, 2^2 + 2^2 = 8, 2^2 = 4, 2^2 + 1^2 + 1^2 = 6, 2^2 + 1^2 = 5, 1^2 + 1^2 + 1^2 + 1^2 = 4, 1^2 + 1^2 + 1^2 = 3, 1^2 + 1^2 = 2, and 1^2 = 1, which are the entries of the 4th row.
The triangle begins:
1
1, 2, 4
1, 2, 3, 5, 9
1, 2, 3, 4, 5, 6, 8, 10, 16
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 17, 25
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 18, 20, 26, 36
...
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MAPLE
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sum_sq:=proc(lst) return add(lst[i]^2, i=1..nops(lst)): end: tot_sum_sq:=proc(lst, tmp_res, strt) local j, new_lst, new_res: new_res:={op(tmp_res), sum_sq(lst)}: for j from strt to nops(lst) do if(lst[j-1]=lst[j] and not (j>2 and lst[j-2]=lst[j]))then new_res:={op(new_res), op(tot_sum_sq(subsop(j=NULL, lst), new_res, j))}: fi: od: return new_res: end: nth_row:=proc(n) local part, parts, res: parts:=combinat[partition](n): res:=[]: for part in parts do res:={op(res), op(tot_sum_sq(part, [], 2))}: od: return res: end: seq(op(nth_row(n)), n=1..7);
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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