Thanks to David E Speyer for the helpful pointers.

I can now answer the first question (is the existence of the function $d$ sufficient?) in the negative.

A function $d$ with the given properties is an isotone valuation as defined in point 6, Chapter V of Birkhoff's "Lattice theory". As in Theorem 9 in the same chapter, we can take a quotient of the lattice, on which the induced $d$ is strictly increasing (called a metric lattice in Birkhoff), and which is therefore a modular lattice.

So the existence of such a $d$ (modulo some finiteness condition) is equivalent to having a non-trivial modular quotient.

Based on this, we might suspect that the existence of such a $d$ is not sufficient to have a non-trivial linear representation, since sublattices of $\text{Sp}(V)$ have further properties (e.g. being arguesian).

As a concrete counter-example, we can take the incidence lattice $L$ of a finite non-Desarguean projective plane. Such an $L$ is modular (the function $d$ can be taken to be $d(point) = 1$, $d(line) = 2$), non-arguesean, and as it turns out also simple (i.e. without any smaller non-trivial quotients). It therefore cannot have a non-trivial arguesean quotient.

The fact that such a lattice $L$ is simple follows from Corollary 1 in Chapter V of Birkhoff:

A modular lattice of finite length is ‘‘simple’’ (i.e., without proper congruence relations) if and only if all its prime quotients are projective.

I believe the fact that there's a line passing through any pair of points is sufficient in this case to show that all prime quotients are projective.

Birkhoff, Garrett, Lattice theory, New York: American Mathematical Society (AMS). v, 155 p. (1940). ZBL0063.00402.

Thanks to David E Speyer for the helpful pointers.

I can now answer the first question (is the existence of the function $d$ sufficient?) in the negative.

A function $d$ with the given properties is an isotone valuation as defined in point 6, Chapter V of Birkhoff's "Lattice theory". As in Theorem 9 in the same chapter, we can take a quotient of the lattice, on which the induced $d$ is strictly increasing (called a metric lattice in Birkhoff), and which is therefore a modular lattice.

So the existence of such a $d$ (modulo some finiteness condition) is equivalent to having a non-trivial modular quotient.

Based on this, we might suspect that the existence of such a $d$ is not sufficient to have a non-trivial linear representation, since sublattices of $\text{Sp}(V)$ have further properties (e.g. being arguesian).

As a concrete counter-example, we can take the incidence lattice $L$ of a finite non-Desarguesian projective plane. Such an $L$ is modular (the function $d$ can be taken to be $d(point) = 1$, $d(line) = 2$), non-arguesean, and as it turns out also simple (i.e. without any smaller non-trivial quotients). It therefore cannot have a non-trivial arguesean quotient.

The fact that such a lattice $L$ is simple follows from Corollary 1 in Chapter V of Birkhoff:

A modular lattice of finite length is ‘‘simple’’ (i.e., without proper congruence relations) if and only if all its prime quotients are projective.

I believe the fact that there's a line passing through any pair of points is sufficient in this case to show that all prime quotients are projective.

Birkhoff, Garrett, Lattice theory, New York: American Mathematical Society (AMS). v, 155 p. (1940). ZBL0063.00402.

Thanks to David E Speyer for the helpful pointers.

I can now answer the first question (is the existence of the function $d$ sufficient?) in the negative.

A function $d$ with the given properties is an isotone valuation as defined in point 6, Chapter V of [1]. As in Theorem 9 in the same chapter, we can take a quotient of the lattice, on which the induced $d$ is strictly increasing (called a metric lattice in Birkhoff), and which is therefore a modular lattice.

So the existence of such a $d$ (modulo some finiteness condition) is equivalent to having a non-trivial modular quotient.

Based on this, we might suspect that the existence of such a $d$ is not sufficient to have a non-trivial linear representation, since sublattices of $\text{Sp}(V)$ have further properties (e.g. being arguesian).

As a concrete counter-example, we can take the incidence lattice $L$ of a finite non-Desarguean projective plane. Such an $L$ is modular (the function $d$ can be taken to be $d(point) = 1$, $d(line) = 2$), non-arguesean, and as it turns out also simple (i.e. without any smaller non-trivial quotients). It therefore cannot have a non-trivial arguesean quotient.

The fact that such a lattice $L$ is simple follows from Corollary 1 in Chapter V of Birkhoff:

A modular lattice of finite length is ‘‘simple’’ (i.e., without proper congruence relations) if and only if all its prime quotients are projective.

I believe the fact that there's a line passing through any pair of points is sufficient in this case to show that all prime quotients are projective.

[1] Birkhoff, Garrett, Lattice theory, New York: American Mathematical Society (AMS). v, 155 p. (1940). ZBL0063.00402.

Thanks to David E Speyer for the helpful pointers.

I can now answer the first question (is the existence of the function $d$ sufficient?) in the negative.

A function $d$ with the given properties is an isotone valuation as defined in point 6, Chapter V of Birkhoff's "Lattice theory". As in Theorem 9 in the same chapter, we can take a quotient of the lattice, on which the induced $d$ is strictly increasing (called a metric lattice in Birkhoff), and which is therefore a modular lattice.

So the existence of such a $d$ (modulo some finiteness condition) is equivalent to having a non-trivial modular quotient.

Based on this, we might suspect that the existence of such a $d$ is not sufficient to have a non-trivial linear representation, since sublattices of $\text{Sp}(V)$ have further properties (e.g. being arguesian).

As a concrete counter-example, we can take the incidence lattice $L$ of a finite non-Desarguean projective plane. Such an $L$ is modular (the function $d$ can be taken to be $d(point) = 1$, $d(line) = 2$), non-arguesean, and as it turns out also simple (i.e. without any smaller non-trivial quotients). It therefore cannot have a non-trivial arguesean quotient.

The fact that such a lattice $L$ is simple follows from Corollary 1 in Chapter V of Birkhoff:

A modular lattice of finite length is ‘‘simple’’ (i.e., without proper congruence relations) if and only if all its prime quotients are projective.

I believe the fact that there's a line passing through any pair of points is sufficient in this case to show that all prime quotients are projective.

Birkhoff, Garrett, Lattice theory, New York: American Mathematical Society (AMS). v, 155 p. (1940). ZBL0063.00402.