What is the function of square brackets around table keys in lua?

I came across tables that have square brackets around keys:

local commands_json =
{
    ["request"] = {
        ["application"] = PW_APPLICATION,
        ["push_token"] = deviceToken
    }
}

Can the square brackets be omitted?

It's simply the long form of specifying keys in a table. You can put any value between the [] (except nil. And floating-point NaNs). Whereas without them, you can only use identifiers.

For example, this is a compile error, since "key name" isn't an identifier (due to the space):

tbl =
{
  key name = 5,
}

In contrast, the following example does work:

tbl =
{
  ["key name"] = 5,
}

The following example with quotes and without square brackets is a compile error:

tbl =
{
  "key name" = 5,
}

If Lua sees a naked value like this, it thinks you're trying to add to the array part of the table. That is, it confuses it with:

tbl =
{
  "key name",
}

Which creates a 1-element array, with tbl[1] equal to "key name". By using [], the compiler can easily tell that you meant for something to be a key rather than the value of an array element.

The long form also lets you distinguish between:

local name = "a name";

tbl =
{
  ["name"] = 5,
  [name] = 7,
}

The second part means to evaluate the expression name, the result of which will be the key. So this table has the keys "name" and "a name".