A215905 Triangle read by rows in which row n contains the possible dimensions of unital *-subalgebras of the n X n complex matrices.

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

Offset: 1

Author

Nathaniel Johnston, Aug 25 2012

Keywords

• nonn
• tabf

Comments

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).

Row lengths are given by A215909.

Examples

			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
...
		

Links

• William Q. Erickson, The Demazure product extended to biwords, arXiv:2407.13165 [math.CO], 2024. See p. 15.

Crossrefs

Cf. A215909, A215914.

Programs

Maple

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);