Monk's formula

Key takeaways

Monk's formula belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Monk's formula to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Monk's formula from memory before moving on to harder problems.

Reference excerpt

In mathematics, Monk's formula, found by Monk (1959), is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold. Write tij for the transposition (i j), and si = ti,i+1. Then 𝔖sr = x1 + ⋯ + xr, and Monk's formula states that for a permutation w,

S s r S w = ∑ i ≤ r < j ℓ ( w t i j ) = ℓ ( w ) + 1 S w t i j , {\displaystyle {\mathfrak {S}}_{s_{r}}{\mathfrak {S}}_{w}=\sum _{{i\leq r<j} \atop {\ell (wt_{ij})=\ell (w)+1}}{\mathfrak {S}}_{wt_{ij}},}

where ℓ ( w ) {\displaystyle \ell (w)} is the length of w. The pairs (i, j) appearing in the sum are exactly those such that i ≤ r < j, wi < wj, and there is no i < k < j with wi < wk < wj; each wtij is a cover of w in Bruhat order.

References Monk, D. (1959), "The geometry of flag manifolds", Proceedings of the London Mathematical Society, Third Series, 9 (2): 253–286, CiteSeerX 10.1.1.1033.7188, doi:10.1112/plms/s3-9.2.253, ISSN 0024-6115, MR 0106911 {{citation}}: Cite uses deprecated parameter |citeseerx= (help)

Worked examples

Example 1 — a first encounter with Monk's formula

Start with the simplest possible case. Write down what Monk's formula claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Monk's formula before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Monk's formula ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Monk's formula

In research: Monk's formula appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Monk's formula in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Monk's formula is common in secondary-school and first-year university syllabi. It links to neighbouring topics Symmetric functions, so understanding it makes those chapters shorter.
In everyday life: Look for Monk's formula outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Monk's formula in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Monk's formula means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Monk's formula out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Monk's formula in simple terms?

In mathematics, Monk's formula, found by Monk (1959), is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold.

Why does Monk's formula matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Monk's formula?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Monk's formula.