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Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true for $|G:H|<32$, because it is true if $[H,G]$ is group-complemented or if it is of rank $2$ (see this paper Corollary 4.33). It's also true (using the function is_cohen_macaulay on SAGE) for the three first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed herehere (my desktop isn't enough powerful for checking the fourth one).

Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true for $|G:H|<32$, because it is true if $[H,G]$ is group-complemented or if it is of rank $2$ (see this paper Corollary 4.33). It's also true (using the function is_cohen_macaulay on SAGE) for the three first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here (my desktop isn't enough powerful for checking the fourth one).

Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true for $|G:H|<32$, because it is true if $[H,G]$ is group-complemented or if it is of rank $2$ (see this paper Corollary 4.33). It's also true (using the function is_cohen_macaulay on SAGE) for the three first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here (my desktop isn't enough powerful for checking the fourth one).

SAGE function is_cohen_macaulay
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Sebastien Palcoux
Sebastien Palcoux
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Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true iffor $|G:H|<32$, because it is true if $[H,G]$ is group-complemented or if it is of rank $2$ (see Corollary 4.33 of this paper Corollary 4.33). It's also true (using the function is_cohen_macaulay on SAGE) for the fourthree first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here (my desktop isn't enough powerful for checking the fourth one).

Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true if $|G:H|<32$ (see Corollary 4.33 of this paper). It's also true for the four first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here.

Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true for $|G:H|<32$, because it is true if $[H,G]$ is group-complemented or if it is of rank $2$ (see this paper Corollary 4.33). It's also true (using the function is_cohen_macaulay on SAGE) for the three first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here (my desktop isn't enough powerful for checking the fourth one).

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Sebastien Palcoux
Sebastien Palcoux
  • 27k
  • 5
  • 74
  • 186

Is the bounded coset poset of a boolean interval of finite groups, Cohen-Macaulay?

Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above by $G$).

Question: Is $\hat{C}(H,G)$ Cohen-Macaulay?

Remark: It is true if $|G:H|<32$ (see Corollary 4.33 of this paper). It's also true for the four first rank $3$ boolean intervals $[H,G]$ with $G$ simple, listed here.