Identification of equivariant maps

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Let $C_n$ be a cyclic group of order $n$ and $X$ be a finite $C_n$-set such that the fixed point set $X^C_r = X$ for the subgroup $C_r$ of $C_n.$ With this setting in hand, I want to understand the mapping space $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X])$. Here $Map^C_n(S,T)$ stands for the set of all $C_n$-equivariant maps from $S$ to $T$ and $mathbbZ[X]$ means the Abelian group generated by the finite set $X.$

If $f : mathbbZ[C_n/C_k] to mathbbZ[X]$ be a $C_n$-equivariant map, then it should factor through the orbit space $mathbbZ[C_n/C_k]/C_r$ as $X^C_r = X.$ Therefore we get a $C_n/C_r$-equivariant map from $mathbbZ[C_n/C_k]/C_r$ to $mathbbZ[X].$

Can we identify the space $mathbbZ[C_n/C_k]/C_r$ with $mathbbZ[C_fracnr/ C_(fracnr,k)]?$

My guess is $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X]) cong Map^C_n/C_r(mathbbZ[C_fracnr/ C_(fracnr,k)], mathbbZ[X]).$ Where $(fracnr,k)$ stands for g.c.d.

Any help will be appreciated.

Thank you so much in advance.

group-actions equivariant-maps

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edited Jul 29 at 12:20

asked Jul 29 at 12:13

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Let $C_n$ be a cyclic group of order $n$ and $X$ be a finite $C_n$-set such that the fixed point set $X^C_r = X$ for the subgroup $C_r$ of $C_n.$ With this setting in hand, I want to understand the mapping space $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X])$. Here $Map^C_n(S,T)$ stands for the set of all $C_n$-equivariant maps from $S$ to $T$ and $mathbbZ[X]$ means the Abelian group generated by the finite set $X.$

If $f : mathbbZ[C_n/C_k] to mathbbZ[X]$ be a $C_n$-equivariant map, then it should factor through the orbit space $mathbbZ[C_n/C_k]/C_r$ as $X^C_r = X.$ Therefore we get a $C_n/C_r$-equivariant map from $mathbbZ[C_n/C_k]/C_r$ to $mathbbZ[X].$

Can we identify the space $mathbbZ[C_n/C_k]/C_r$ with $mathbbZ[C_fracnr/ C_(fracnr,k)]?$

My guess is $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X]) cong Map^C_n/C_r(mathbbZ[C_fracnr/ C_(fracnr,k)], mathbbZ[X]).$ Where $(fracnr,k)$ stands for g.c.d.

Any help will be appreciated.

Thank you so much in advance.

group-actions equivariant-maps

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edited Jul 29 at 12:20

asked Jul 29 at 12:13

Math

35819

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Let $C_n$ be a cyclic group of order $n$ and $X$ be a finite $C_n$-set such that the fixed point set $X^C_r = X$ for the subgroup $C_r$ of $C_n.$ With this setting in hand, I want to understand the mapping space $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X])$. Here $Map^C_n(S,T)$ stands for the set of all $C_n$-equivariant maps from $S$ to $T$ and $mathbbZ[X]$ means the Abelian group generated by the finite set $X.$

If $f : mathbbZ[C_n/C_k] to mathbbZ[X]$ be a $C_n$-equivariant map, then it should factor through the orbit space $mathbbZ[C_n/C_k]/C_r$ as $X^C_r = X.$ Therefore we get a $C_n/C_r$-equivariant map from $mathbbZ[C_n/C_k]/C_r$ to $mathbbZ[X].$

Can we identify the space $mathbbZ[C_n/C_k]/C_r$ with $mathbbZ[C_fracnr/ C_(fracnr,k)]?$

My guess is $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X]) cong Map^C_n/C_r(mathbbZ[C_fracnr/ C_(fracnr,k)], mathbbZ[X]).$ Where $(fracnr,k)$ stands for g.c.d.

Any help will be appreciated.

Thank you so much in advance.

group-actions equivariant-maps

share|cite|improve this question

edited Jul 29 at 12:20

asked Jul 29 at 12:13

Math

35819

Let $C_n$ be a cyclic group of order $n$ and $X$ be a finite $C_n$-set such that the fixed point set $X^C_r = X$ for the subgroup $C_r$ of $C_n.$ With this setting in hand, I want to understand the mapping space $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X])$. Here $Map^C_n(S,T)$ stands for the set of all $C_n$-equivariant maps from $S$ to $T$ and $mathbbZ[X]$ means the Abelian group generated by the finite set $X.$

If $f : mathbbZ[C_n/C_k] to mathbbZ[X]$ be a $C_n$-equivariant map, then it should factor through the orbit space $mathbbZ[C_n/C_k]/C_r$ as $X^C_r = X.$ Therefore we get a $C_n/C_r$-equivariant map from $mathbbZ[C_n/C_k]/C_r$ to $mathbbZ[X].$

Can we identify the space $mathbbZ[C_n/C_k]/C_r$ with $mathbbZ[C_fracnr/ C_(fracnr,k)]?$

My guess is $Map^C_n(mathbbZ[C_n/C_k], mathbbZ[X]) cong Map^C_n/C_r(mathbbZ[C_fracnr/ C_(fracnr,k)], mathbbZ[X]).$ Where $(fracnr,k)$ stands for g.c.d.

Any help will be appreciated.

Thank you so much in advance.

group-actions equivariant-maps

share|cite|improve this question

edited Jul 29 at 12:20

asked Jul 29 at 12:13

Math

35819

share|cite|improve this question

share|cite|improve this question

edited Jul 29 at 12:20

edited Jul 29 at 12:20

edited Jul 29 at 12:20

asked Jul 29 at 12:13

Math

35819

asked Jul 29 at 12:13

Math

35819

asked Jul 29 at 12:13

Math

35819

35819

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