The determinant of a matrix is given by the well-known formula(adsbygoogle = window.adsbygoogle || []).push({});

det(A) = sum_{p}parity(p) * product_{i = 1...n}A_{i,p(i)}

where the p's are all permutations of 1...n and A is a n*n matrix. Parity is +1 for an even permutation and -1 for an odd one.

For a permanent, replace parity(p) with 1. For an immanant, replace parity(p) with X(q,p), a character of the symmetric group for irreducible representation q.

Determinants show up in many places that matrices do. Permanents are the bosonic counterpart of the fermionic Slater determinant of multiplarticle wavefunctions. Immanants appear in the theory of Schur functions. But I've seen much less of permanents and immanants than determinants. Could that be due to not having certain nice properties that determinants have?

In particular, determinants satisfy the multiplication law det(A.B) = det(A)*det(B) For permanents or immanants, however, one can find counterexamples. Is that law true for any generalization other than determinants themselves?

Let's define a generalized determinant gdt(A) = sum_{p}X(p) * product_{i = 1...n}A_{i,p(i)}

and let's set gdt(A.B) = gdt(A)*gdt(B) for all possible A and B. What constraints can one find on the X(p)'s? Can one show that only X(p) = parity(p) will yield that multiplication law?

So far, I've found that all p's where X(p) is nonzero must form a group, and that these nonzero X(p)'s with multiplication form an abelian quotient group of the group of p's. That includes determinants and permanents, and gdt's for various other groups of p's.

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# Generalizations of Determinants: Permanents, Immanants, etc.

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