Examples of improved notation that impacted research?

The notation that I found extremely useful to support my intuition is from Grothendieck $begin{array} \{cal F} \ hspace{0.05in}mid \X end{array}$ to denote a sheaf $cal F$ over a topological space $X$.

There is a notation that had an immediate and profound impact on research in algebraic topology, later algebraic geometry, and was eventually adopted by all areas of mathematics: the introduction of arrows to denote mappings. Compare $f colon X to Y$ with $f(X) subset Y$, which is what was used previously. It meets all three criteria mentioned by the OP and is recognized by every mathematician.

Just as importantly, the use of arrows led to commutative diagrams, without which many parts of modern mathematics are now inconceivable. I mentioned this before in an answer here.

This might be too old to qualify but I have always felt that the decimal notation is a wonderful thing!

From a modern perspective, when it is taught to everyone at a very young age, it might be hard to appreciate how interesting and useful it is. The fact that it is at all possible relies on the following simple but non trivial theorems: $$sum_{k=0}^n(a-1)a^k = a^{n+1}-1$$ and for $|x_k| leq a-1$ $$sum_{kleq n}x_k a^k = 0 implies x_k =0.$$

The notation is, in fact, mathematically optimal for representing numbers in terms of information theory!

I think it is a hallmark of the greatest notations to encode non trivial theorems so that they seem like trivialities. This leads to great cognitive savings in my experience.

It is very clear that it led to vast improvements, not only in mathematical research but human society as a whole.

What might be called the Gelfand Philosophy notation has become popular in the field of 'Woronowicz' quantum groups in the past decade.

The idea starts with the Gelfand theorem that a commutative $mathrm{C}^*$-algebra $A$ is isometrically isomorphic to $C_0(X)$, for $X$ a particular topological space, certainly compact and Hausdorff if $A$ is unital, in which case $Acong C(X)$. Restricting now to the unital case, this Gelfand Philosophy says that a noncommutative $mathrm{C}^*$-algebra $A$ should be thought of as the algebra of continuous functions on a compact quantum space, $mathbb{X}$, and so we write $A=:C(mathbb{X})$. Of course $mathbb{X}$ is not a set, let alone a topological space but a so-called virtual object. A more radical (if only stylistically) approach is not to use blackboard bold to signify that $mathbb{X}$ is a virtual object but just to use $X$.

For examples of where this goes I want to talk about compact quantum groups. Compact quantum groups are spoken about through what are called algebras of functions on the compact quantum group. For example, a compact quantum group $G$ might be spoken about via it algebra of continuous functions $C(G)$, a (Woronowicz) $mathrm{C}^*$-algebra. These algebra of functions have Haar states $h$ that are precisely integration against the Haar measure whenever $C(G)$ is commutative/$G$ is classical. Playing a little fast and loose with issues of null sets, in the classical case we can define $$mathcal{L}^2(G)=left{fin C(G)mid int_G |f(t)|^2,dmu(t)<inftyright},$$ and via $|f|:=f^*f$ for $fin C(G)$ non-commutative, we can also define $mathcal{L}^2(G)$ spaces for compact quantum groups $G$: $$mathcal{L}^2(G)=left{fin C(G)mid h(|f|^2)<inftyright}.$$ This kind of thing can go in all kinds of directions, the basic principle is if you have a notation for something in commutative algebras of functions/classical groups that makes sense for noncommutative algebra of functions/quantum groups, use that same notation for quantum groups.

This also allows you to, in a strictly nonsensical but useful way, to talk about the quantum group as if it really exists. For example for finite groups at least, with full algebra of functions $F(G)$, there is a bijective correspondence with representation $Grightarrow L(V)$ and corepresentations $Vrightarrow Votimes F(G)$. Through this lens one can talk about a representation of a quantum group or in a similar way the action of a quantum group.

To actually answer the question asked I will quote from a recent preprint:

When, for example with the representation theory of compact quantum groups, the noncommutative theory generalises so nicely from the commutative theory, it can be useful to refer to a virtual object as if it exists: this approach helps point towards appropriate noncommutative definitions, and sometimes even towards results, such as the Peter-Weyl Theorem, that are true in this larger class of objects. Even when commutative results do not generalise to this larger class, the Gelfand Philosophy gives a pleasing notation, helping readers from the commutative world understand better what is going on in the noncommutative world.

Some examples of this from my own work on finite quantum groups, say given by a $mathrm{C}^*$-algebra $A$ include:

• referring to $A$ as the algebra of functions on the finite quantum group $G$, and denote it by $F(G)$
• referring to the unit $1_A$ in the algebra of functions as $mathbf{1}_G$
• referring to the set of states $mathcal{S}(A)$ as $M_p(G)$, the set of probability measures on the group
• I have started using $fin F(G)$ for a general "function" rather than the usual $ain F(G)$ or before $ain A$
• I have used the notation $2^G$ for the set of projections in $F(G)$

Some of these are pushing the envelope a little on this notation... we will see what the Reviewers say!

I note in this preprint that:

This philosophical approach ramped up in the 2000s, and into the 2010s, and up to 2020.

HOWEVER, when I got my hand on the 1967 paper of Kac and Paljutkin (highly recommended if you can get a copy), the famous eight-dimensional algebra of functions on a finite quantum group, the smallest of which is neither commutative nor cocommutative, the authors refer to it by $mathfrak{G}_0$ --- not the algebra but the virtual object!

I assume similar notations are at play in other fields.

I like to use figures to represent quantities. For example, in this recent preprint, my coauthor and I use simple diagrams to represent certain weighted sums (polynomials). Writing out the sums explicitly would be extremely cumbersome to parse, and any sane reader would just convert it back to a generic figure anyway, in order to understand the sum.

Richard Stanley’s symbol for number of ways to make choices with replacement. Looks like a binomial coefficient but with double parentheses. (More in my blog.)

It's a calculation that comes up frequently -- I became more aware of just how frequently it comes up when I started giving it it's own symbol -- and it helps to give it its own notation, even though it reduces to a simple expression in terms of binomial coefficients.

May I offer the four-vector notation as an example from physics? Quoting Feynman:

The notation for four-vectors is different than it is for three- vectors. [...] We write $p_mu$ for the four-vector, and $mu$ stands for the four possible directions $t$, $x$, $y$, or $z$. We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations. The whole idea of a four- vector, in fact, is an improvement in notation so that the transformations can be remembered easily.

The Feynman Lecture on Physics, Volume 1, Chapter 17.

A trigonometric notation that Feynman invented in his youth did not catch on. And then of course Feynman diagrams are perhaps the most celebrated example of an impactful notation in physics.