Specifically, if you were one of the creatures given permission to move in and out of the Hut, can you cast a concentration spell from outside the Hut, then step inside it and maintain concentration?
1 Answer
The dome is not a barrier to concentration
Tiny hut says:
Creatures and objects within the dome when you cast this spell can move through it freely. All other creatures and objects are barred from passing through it. Spells and other magical effects can't extend through the dome or be cast through it.
Based on this, you might think that the dome blocks concentration, either because 'the dome generally blocks things' or because concentration is explicitly a magical effect.
The first reason is easier to address. Spells do (only) what they say they do, and tiny hut says that it wards against creatures, objects, spells, and magical effects. It also implicitly blocks uncomfortable and wet weather, and blocks light leaving but not entering. It does not say that it blocks concentration, so unless concentration is a creature, object, spell, or magical effect (or weather or light), it is not blocked. Consider that the hut does not block a dragon's breath weapon, or sound, because these are not one of the things it explicitly or implicitly blocks.
The dome of the hut is a barrier and does block line of sight from the outside, but concentration is not blocked by barriers:
Sage Advice tells us:
If you’re concentrating on a spell, do you need to maintain line of sight with the spell’s target or the spell’s effect? You don’t need to be within line of sight or within range to maintain concentration on a spell, unless a spell’s description or other game feature says otherwise.
So, the only way that tiny hut would block concentration was if concentration was a magical effect. Is it?
Casting a Spell (PHB):
Each spell description begins with a block of information, including the spell's name, level, school of magic, casting time, range, components, and duration. The rest of a spell entry describes the spell's effect.
Duration (PHB):
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
If a spell must be maintained with concentration, that fact appears in its Duration entry, and the spell specifies how long you can concentrate on it.
So, concentration is not part of a spell's effect. Rather, it is part of a spell's duration, which is itself information about the spell. The dome of the tiny hut blocks magical effects from passing through, but it does not block 'spell duration'. This makes sense, because you would not expect that, for example, a non-concentration spell or other effect with a duration of 1 hour would have the duration of its effect either cut short or extended by the dome. For example, a creature under the effect of a beholder's 1-hour charm would not have this charm ended by entering the dome.