Here's a problem that I believe has little chance of being resolved, and it's also not so clear to me what the motivation behind the problem is, but it involves some very pretty algebraic combinatorics.

In "Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations", Proctor defines a poset $P$ to be Gaussian if for every $m\geq 0$ we have $$ \sum_{I \in J(P\times [m])} q^{\#I} = \frac{\prod_{i=1}^{r}(1-q^{h_i+m})}{\prod_{i=1}^{r}(1-q^{h_i})}$$ where $r$ and $h_1,\ldots,h_r$ are constants independent of $m$. (Here $J(P \times [m])$ is the distributive lattice of order ideals of the Cartesian product of $P$ and the chain poset $[m]=\{1,2,\ldots,m\}$.) Using results from Standard Monomial Theory, he proves that if $P$ is a minuscule poset, then $P$ is Gaussian. Here a minuscule poset is a certain finite poset coming from a minuscule representation of a semisimple Lie algebra; these have been classified- see Proctor's paper for the list. Furthermore, Proctor conjectures that the minuscule posets are the only Gaussian posets.

I think a few things are known (due to some mixture of Proctor and Stanley) about this conjecture: that a Gaussian poset $P$ is a disjoint union of connected Gaussian posets; that a connected Gaussian poset has to be graded; that $r$ has to be the number of elements of $P$; that the $h_i$ have to be the ranks of the elements $p\in P$; maybe some more things too. But like I said I think the full conjecture is pretty hopeless.

Here's a problem that I believe has little chance of being resolved, and it's also not so clear to me what the motivation behind the problem is, but it involves some very pretty algebraic combinatorics.

Stanley (in his PhD thesis) defined a poset $P$ to be Gaussian if for every $m\geq 0$ we have $$ \sum_{I \in J(P\times [m])} q^{\#I} = \frac{\prod_{i=1}^{r}(1-q^{h_i+m})}{\prod_{i=1}^{r}(1-q^{h_i})}$$ where $r$ and $h_1,\ldots,h_r$ are constants independent of $m$. (Here $J(P \times [m])$ is the distributive lattice of order ideals of the Cartesian product of $P$ and the chain poset $[m]=\{1,2,\ldots,m\}$.) 

Using results from Standard Monomial Theory, in "Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations", Proctor proved that if $P$ is a minuscule poset, then $P$ is Gaussian. Here a minuscule poset is a certain finite poset coming from a minuscule representation of a semisimple Lie algebra; these have been classified- see Proctor's paper for the list. Furthermore, Proctor conjectured that the minuscule posets are the only Gaussian posets.

I think a few things are known (due to some mixture of Proctor and Stanley) about this conjecture: that a Gaussian poset $P$ is a disjoint union of connected Gaussian posets; that a connected Gaussian poset has to be graded; that $r$ has to be the number of elements of $P$; that the $h_i$ have to be the ranks of the elements $p\in P$; maybe some more things too. But like I said I think the full conjecture is pretty hopeless.

Here's a problem that I believe has little chance of being resolved, and it's also not so clear to me what the motivation behind the problem is, but it involves some very pretty algebraic combinatorics.

In "Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations", Proctor defines a poset $P$ to be Gaussian if for every $m\geq 0$ we have $$ \sum_{I \in J(P\times [m])} q^{\#T} = \frac{\prod_{i=1}^{r}(1-q^{h_i+m})}{\prod_{i=1}^{r}(1-q^{h_i})}$$ where $r$ and $h_1,\ldots,h_r$ are constants independent of $m$. (Here $J(P \times [m])$ is the distributive lattice of order ideals of the Cartesian product of $P$ and the chain poset $[m]=\{1,2,\ldots,m\}$.) Using results from Standard Monomial Theory, he proves that if $P$ is a minuscule poset, then $P$ is Gaussian. Here a minuscule poset is a certain finite poset coming from a minuscule representation of a semisimple Lie algebra; these have been classified- see Proctor's paper for the list. Furthermore, Proctor conjectures that the minuscule posets are the only Gaussian posets.

I think a few things are known (due to some mixture of Proctor and Stanley) about this conjecture: that the poset $P$ in question has to be graded; that $r$ has to be the number of elements of $P$; that the $h_i$ have to be the ranks of the elements $p\in P$; maybe some more things too. But like I said I think the full conjecture is pretty hopeless.

Here's a problem that I believe has little chance of being resolved, and it's also not so clear to me what the motivation behind the problem is, but it involves some very pretty algebraic combinatorics.

In "Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations", Proctor defines a poset $P$ to be Gaussian if for every $m\geq 0$ we have $$ \sum_{I \in J(P\times [m])} q^{\#I} = \frac{\prod_{i=1}^{r}(1-q^{h_i+m})}{\prod_{i=1}^{r}(1-q^{h_i})}$$ where $r$ and $h_1,\ldots,h_r$ are constants independent of $m$. (Here $J(P \times [m])$ is the distributive lattice of order ideals of the Cartesian product of $P$ and the chain poset $[m]=\{1,2,\ldots,m\}$.) Using results from Standard Monomial Theory, he proves that if $P$ is a minuscule poset, then $P$ is Gaussian. Here a minuscule poset is a certain finite poset coming from a minuscule representation of a semisimple Lie algebra; these have been classified- see Proctor's paper for the list. Furthermore, Proctor conjectures that the minuscule posets are the only Gaussian posets.

I think a few things are known (due to some mixture of Proctor and Stanley) about this conjecture: that a Gaussian poset $P$ is a disjoint union of connected Gaussian posets; that a connected Gaussian poset has to be graded; that $r$ has to be the number of elements of $P$; that the $h_i$ have to be the ranks of the elements $p\in P$; maybe some more things too. But like I said I think the full conjecture is pretty hopeless.