Immanants

Recently computer scientist Leslie Valliant won the ACM’s Turing Award, considered one of the most prestigious prizes in computer science. Valliant is famous for many results, not the least of which are his results on the Permanent of a matrix. Over at the Godel’s Lost Letter, the iced tea man has a nice collection of interesting permanent related complexity facts. Recall that the permanent of a n by n matrix [latex]A[/latex] is given by

[latex]{rm per} A = sum_{pi in S_n} prod_{i=1}^n A_{i,pi(i)}[/latex]

where [latex]S_n[/latex] is the symmetric group on n elements and similarly the determinant of a n by n matrix [latex]A[/latex] is given by

[latex]{rm det} A = sum_{pi in S_n} prod_{i=1}^n (-1)^{{rm sgn} pi} A_{i,pi(i)}[/latex]

where [latex]{rm sgn} pi[/latex] is 0 if the permutation is made up of an even number of transpositions and 1 if the permutation is made up of an odd number of transpositions. One day I was sitting in my office when a physics undergraduate came by (a former ex-sailor from Alaska) and said…”Hey Dave, what if we replace the function in front of each term in the permanent and determinant by a character of a symmetric group irrep?” Which of course knocked me off my chair, first because what undergrad physics major knows about symmetric group irreps and second because I had never thought about this interesting twist on the permanent and determinant.
After a little google foo later, we quickly found the answer. For an n by n matrix [latex]A[/latex] the immanant of a matrix is given by

[latex]{rm imm_lambda A } = sum_{pi in S_n} prod_{i=1}^n chi_{lambda}(pi) A_{i,pi(i)}[/latex]

where [latex]lambda[/latex] labels the irrep of [latex]S_n[/latex] and [latex]chi_lambda(pi)[/latex] is the character of the irrep [latex]lambda[/latex] at group element [latex]pi[/latex]. Recall that the irreps of the symmetric group [latex]S_n[/latex] are labeled by partitions of [latex]n[/latex]. A partition of [latex]n[/latex] is a series of decreasing positive integers that sums to n, [latex](lambda_1m lambda_2, dots,lambda_r)[/latex] with [latex]lambda_1 geq lambda_2 geq dots geq lambda_r[/latex] such that [latex]sum_{i=1}^r lambda_i = n[/latex]. The partition corresponding to [latex](n)[/latex] corresponds to the trivial irrep in which [latex]chi_{(n)}(pi)=1[/latex], and on the opposite end of the spectrum, the partition corresponding to [latex](1,1,dots,1)[/latex] corresponds to the alternating irrep where [latex]chi_{(1,1,dots,1)}(pi)=(-1)^{{rm sgn} pi}[/latex]. Thus we see that the permanent and determinant are at the end of a spectrum of polynomials known as the immanants.
One of Valiant’s most well known results is that evaluating the permanent of a matrix with 0 and 1 as its possible matrix entries is #P complete, a.k.a really hard. On the other hand evaluating the determinant is not computationally difficult at all. At first this seems odd because a determinant has those minus signs which you would think would make it easier and not hard, but alas this is not so. So what about the computational complexity of the immanant? The Computational Complexity of Immanants by Peter Bürgisser (sorry I could only find a paywalled version) shows that there is a since in which certain immanants are also #P complete to evaluate. Rough the idea is that if one defines a class of immanants that have partitions that have a “step” in them that grows polynomially in [latex]n[/latex] (the step for the permanent is [latex]n[/latex]) then these will be #P complete to evaluate.
So what good are immanants? Well I’m sure mathematicians have some fine uses for them. One interesting thing to note is that immanantal polynomials of adjacency matrices of graphs give you graph invariants (for the determinant this is the same as saying that the eigenvalues of the adjacency matrix are a graph invariant.) However it is known that this, like the eigenvalues, is not a complete set of graph invariants and so is not a route towards efficiently solving graph isomorphism. So no real use there, but I’m sure an object so symmetric must be of some use, no?