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What is the name of this generalization of permanent and determinant?
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Let $G$ be a subgroup of $S_n$, we view the elements as bijective functions of $1, ldots, n$ to itself. Let $chi : G to mathbbC$ be a character (group homomorphism, one-dimensional representation). Then there is a map,
$$Delta_G, chi colon Mat(n, mathbbC) to mathbbC$$
Given by:
$$Delta_G, chi ((a_i,j)_i, j = 1^n) = sum_g in G chi(g)prod_i=1^n a_i, g(i)$$
When $G = S_n$ (the full group) there are only two choices of $chi$: the trivial representation and the sign-representation. In these cases the definition above reproduces the standard definition of the permanent and determinant respectively.
It would seem to me that this more general class of permanent/determinant-like function has been studied before, e.g. in the 19th century when determinants were really popular. Do you know if has a name?
representation-theory terminology determinant
edited Jul 19 at 13:44
joriki
164k10180328
asked Jul 19 at 12:43
Vincent
2,720925
add a comment |Â
up vote
3
down vote
favorite
Let $G$ be a subgroup of $S_n$, we view the elements as bijective functions of $1, ldots, n$ to itself. Let $chi : G to mathbbC$ be a character (group homomorphism, one-dimensional representation). Then there is a map,
$$Delta_G, chi colon Mat(n, mathbbC) to mathbbC$$
Given by:
$$Delta_G, chi ((a_i,j)_i, j = 1^n) = sum_g in G chi(g)prod_i=1^n a_i, g(i)$$
When $G = S_n$ (the full group) there are only two choices of $chi$: the trivial representation and the sign-representation. In these cases the definition above reproduces the standard definition of the permanent and determinant respectively.
It would seem to me that this more general class of permanent/determinant-like function has been studied before, e.g. in the 19th century when determinants were really popular. Do you know if has a name?
representation-theory terminology determinant
edited Jul 19 at 13:44
joriki
164k10180328
asked Jul 19 at 12:43
Vincent
2,720925
3
Immanent$$.
– Lord Shark the Unknown
Jul 19 at 12:47
4
en.wikipedia.org/wiki/Immanant
– Chappers
Jul 19 at 12:52
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $G$ be a subgroup of $S_n$, we view the elements as bijective functions of $1, ldots, n$ to itself. Let $chi : G to mathbbC$ be a character (group homomorphism, one-dimensional representation). Then there is a map,
$$Delta_G, chi colon Mat(n, mathbbC) to mathbbC$$
Given by:
$$Delta_G, chi ((a_i,j)_i, j = 1^n) = sum_g in G chi(g)prod_i=1^n a_i, g(i)$$
When $G = S_n$ (the full group) there are only two choices of $chi$: the trivial representation and the sign-representation. In these cases the definition above reproduces the standard definition of the permanent and determinant respectively.
It would seem to me that this more general class of permanent/determinant-like function has been studied before, e.g. in the 19th century when determinants were really popular. Do you know if has a name?
representation-theory terminology determinant
edited Jul 19 at 13:44
joriki
164k10180328
asked Jul 19 at 12:43
Vincent
2,720925
Let $G$ be a subgroup of $S_n$, we view the elements as bijective functions of $1, ldots, n$ to itself. Let $chi : G to mathbbC$ be a character (group homomorphism, one-dimensional representation). Then there is a map,
$$Delta_G, chi colon Mat(n, mathbbC) to mathbbC$$
Given by:
$$Delta_G, chi ((a_i,j)_i, j = 1^n) = sum_g in G chi(g)prod_i=1^n a_i, g(i)$$
When $G = S_n$ (the full group) there are only two choices of $chi$: the trivial representation and the sign-representation. In these cases the definition above reproduces the standard definition of the permanent and determinant respectively.
It would seem to me that this more general class of permanent/determinant-like function has been studied before, e.g. in the 19th century when determinants were really popular. Do you know if has a name?
representation-theory terminology determinant
edited Jul 19 at 13:44
joriki
164k10180328
asked Jul 19 at 12:43
Vincent
2,720925
edited Jul 19 at 13:44
joriki
164k10180328
edited Jul 19 at 13:44
joriki
164k10180328
edited Jul 19 at 13:44
joriki
164k10180328
164k10180328
asked Jul 19 at 12:43
Vincent
2,720925
asked Jul 19 at 12:43
Vincent
2,720925
asked Jul 19 at 12:43
Vincent
2,720925
2,720925
3
Immanent$$.
– Lord Shark the Unknown
Jul 19 at 12:47
4
en.wikipedia.org/wiki/Immanant
– Chappers
Jul 19 at 12:52
add a comment |Â
3
Immanent$$.
– Lord Shark the Unknown
Jul 19 at 12:47
4
en.wikipedia.org/wiki/Immanant
– Chappers
Jul 19 at 12:52
3
3
Immanent$$.
– Lord Shark the Unknown
Jul 19 at 12:47
Immanent$$.
– Lord Shark the Unknown
Jul 19 at 12:47
4
4
en.wikipedia.org/wiki/Immanant
– Chappers
Jul 19 at 12:52
en.wikipedia.org/wiki/Immanant
– Chappers
Jul 19 at 12:52
add a comment |Â
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3
Immanent$$.
– Lord Shark the Unknown
Jul 19 at 12:47
4
en.wikipedia.org/wiki/Immanant
– Chappers
Jul 19 at 12:52