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# 1 Partitions and Rogers-Ramanujan identities

## 1.1 Partitions

• $\sf 16 = 6+3+3+2+1+1$

• Difference at least $\sf something$ at distance $\sf something$
• $\sf 17 = 8+5+3+1$ satisfies "difference at least 2 at distance 1"
• $\sf 76 = 16+13+13+12+11+11$ satisfies "difference at least 2 at distance 3"
• Initial Conditions, which often read like:
• At most a certain number of $\sf 1$'s
• $\sf 1$'s are allowed, but no $\sf 2$'s
• Total number of $\sf 1$'s and $\sf 2$'s can not exceed a certain bound.
• No $\sf 1$'s or $\sf 2$'s allowed

• Congruence conditions: Each part satisfying a congruence modulo something.

## 1.2 Rogers-Ramanujan identities

For any non-negative integer $\sf n$ (and a formal variable $\sf q$),

RR1 : # of partitions with each part $\sf \not\equiv 0,\pm 2 \pmod{5}$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 1$

$\sf \prod\limits_{m>0,\,\, m \,\not\equiv \,0, \pm 2 \pmod{5}} \frac{1}{(1-q^m)} = \sum\limits_{n=0}^\infty d_1(n)q^n$

RR2 : # of partitions with each part $\sf \not\equiv 0,\pm 1 \pmod{5}$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 1$ and with $\sf 1$ appearing at most $\sf0$ times

$\sf \prod\limits_{m>0,\,\, m \,\not\equiv\, 0,\pm 1 \pmod{5}} \frac{1}{(1-q^m)}=\sum\limits_{n=0}^\infty d_2(n)q^n$

# of partitions of type RR1 $\sf \ge$ # of partitions of type RR2.
(Hint: look at the sum side).

Ehrenpreis's question: Can one deduce this without looking at the sum sides?

## 1.3 Examples

RR1 : # of partitions with each part $\sf \not\equiv 0,\pm 2 \pmod{5}$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 1$

 \begin{align*} \sf 10 &= \sf 9+1\\ \sf &= \sf 6+4\\ \sf &= \sf 6+1+1+1+1\\ \sf &= \sf 4+4+1+1\\ \sf &=\sf 4+1+1+1+1+1+1\\ \sf &=\sf 1+1+\cdots+1+1\quad\quad \end{align*} \begin{align*} \sf 10 &=\sf 10\\ &= \sf 9+1\\ \sf &= \sf 8 + 2\\ \sf &= \sf 7 + 3 \\ \sf &= \sf 6+4 \\ \sf & = \sf 6+3+1 \quad\quad \end{align*}

RR2 : # of partitions with each part $\sf \not\equiv 0,\pm 1 \pmod{5}$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 1$ and with $\sf 1$ appearing at most $\sf0$ times

 \begin{align*} \sf 10 &= \sf 8+2\\ \sf &= \sf 7+3\\ \sf &= \sf 3+3+2+2\\ \sf &=\sf 2+2+2+2+2\quad\quad \end{align*} \begin{align*} \sf 10 &=\sf 10\\ \sf &= \sf 8 + 2\\ \sf &= \sf 7 + 3 \\ \sf &= \sf 6+4 \quad\quad \end{align*}

## 1.4 Generalizations

### [i] Gordon-Andrews Identities

• For all odd moduli
• Fix $\sf k>0$
• For any $\sf i=1,\dots,k$,

# of partitions with each part $\sf \not\equiv 0, \pm (k-i+1) \text{ (mod }2k+1)$
$\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf k-1$ and with $\sf 1$ appearing at most $\sf k-i$ times.
• So, RR correspond to $\sf k=2$. (RR1: $\sf k=2, i=1$, RR2: $\sf k=2, i=2$.)

### [ii] Andrews-Bressoud Identities

• For all even moduli
• Analogous product sides
• An added "parity" condition in the sum-side.

### [iii] Others

Goellnitz-Gordon-Andrews, Capparelli, Nandi, some recent conjectures of M. C. Russell and S. K.

## 1.5 Example: Gordon-Andrews

Say $\sf k=4$, hence, there are $\sf 4$ GA identities, the modulus is $\sf 9.$

GA4,1: # of partitions with each part $\sf \not\equiv 0, \pm 4 \text{ (mod } 9)$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 3$ times.

GA4,2: # of partitions with each part $\sf \not\equiv 0, \pm 3 \text{ (mod } 9)$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 2$ times.

GA4,3: # of partitions with each part $\sf \not\equiv 0, \pm 2 \text{ (mod } 9)$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 1$ times.

GA4,4: # of partitions with each part $\sf \not\equiv 0, \pm 1 \text{ (mod } 9)$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 0$ times.

GG4,1 $\sf \ge$ GG4,2 $\sf \ge$ GG4,3 $\sf \ge$ GG4,4.

## 1.6 Example: Andrews-Bressoud

Say $\sf k=4$, hence, there are $\sf 4$ AB identities, the modulus is $\sf 8.$

AB4,1: something related to "$\sf \not\equiv 0, \pm 4 \text{ (mod } 8)$" $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 3$ times + a parity condition.

AB4,2: # of partitions with each part $\sf \not\equiv 0, \pm 3 \text{ (mod } 8)$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 2$ times + a parity condition.

AB4,3: # of partitions with each part $\sf \not\equiv 0, \pm 2 \text{ (mod } 8)$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 1$ times + a parity condition.

AB4,4: # of partitions with each part $\sf \not\equiv 0, \pm 1 \text{ (mod } 8)$ $\sf =$ # of partitions with difference at least $\sf 2$ at distance $\sf 3$ and with $\sf 1$ appearing at most $\sf 0$ times + a parity condition.

AB4,1 $\sf \ge$ AB4,3 and AB4,2 $\sf \ge$ AB4,4.

# 2 Relations to Affine Lie algebras and Vertex Algebras

## 2.1 The work of Lepowsky-Wilson

• If $\sf V = \oplus_{n=0}^\infty V_n$ then the Graded Dimension of $\sf V$ is $\sf \text{dim}_*V = \sum\limits_{n=0}^\infty(\text{dim }V_n) q^n$.
• Consider the "principally graded" "standard modules" for the affine Lie algebra "$\sf A_1^{(1)}$" at a fixed non-negative integral "level" $\sf \ell$.
• There are $\sf\ell+1$ such modules which fall into $\sf k=\left\lceil\frac{\ell+1}{2}\right\rceil$ outer-isomorphism classes. Therefore, there are $\sf k$ graded dimensions to look at.
• Using character formula and Lepowsky-Milne numerator formula:
• If $\sf \ell$ odd: the graded dimensions correspond to the $\sf k$ products in Gordon-Andrews identities (odd modulus)
• If $\sf \ell$ even: the graded dimensions correspond to the $\sf k$ products in Andrews-Bressoud identities (even modulus)
• Therefore, the RR identities live at $\sf \ell=3$ i.e., $\sf k=2$.
• Lepowsky-Wilson explained the sum sides by inventing what they called "$\sf Z$-algebras".
• They gave a $\sf Z$-algebraic proof of a pair of Euler identities $(\sf \ell=2)$ and of RR identities $(\sf \ell=3)$. For the rest of the levels, they gave a $\sf Z$-algebraic interpretation of the corresponding identities.
• One exhibits a basis (or, at least a "small" spanning set) of each of the $\sf V_n$'s, by applying monomials in $\sf Z$-operators applied to a highest weight vector. These monomials satisfy requisite difference and initial conditions. (ex. $\sf Z_{-8}Z_{-5}Z_{-3}Z_{-2}v$)
• This approach treats all levels (regardless of the parity of $\sf \ell$) on an equal footing.

## 2.2 Some examples

 Algebra Level Rogers-Ramanujan $\sf A_1^{(1)}$ $\sf 3$ Gordon-Andrews for modulus $\sf 2k+1$ $\sf A_1^{(1)}$ $\sf 2k-1$ Euler $\sf A_1^{(1)}$ $\sf 2$ Andrews-Bressoud for modulus $\sf 2k$ $\sf A_1^{(1)}$ $\sf 2k-2$ Goellnitz-Gordon (modulus $\sf 8$) $\sf A_5^{(2)}$ $\sf 2$ Capparelli $\sf A_2^{(2)}$ $\sf 3$ Nandi $\sf A_2^{(2)}$ $\sf 4$ M. C. Russell - S. K. mod $\sf 9$ conjectures $\sf D_4^{(3)}$ $\sf 3$

# 3 Motivated Proofs

## 3.1 Andrews-Baxter's motivated proof of RR

Motivated by Ehrenpreis's question (RR1 $\sf \ge$ RR2, without looking at sum-sides)

\sf\begin{align*} G_1 &= \sf \prod\limits_{m>0,\,\, m \,\not\equiv \,0, \pm 2 \pmod{5}} \frac{1}{(1-q^m)} = 1 + q + q^2 + q^3+2q^4+2q^5+3q^6+3q^7+4q^8+5q^9+6q^{10}+\cdots\\ \sf G_2 &= \sf \prod\limits_{m>0,\,\, m \,\not\equiv \,0, \pm 1 \pmod{5}} \frac{1}{(1-q^m)} = 1 +\hphantom{q}+q^2+q^3+\,\,q^4+\,\,q^5\,+2q^6+2q^7+3q^8+4q^9+4q^{10}+\cdots\\ \end{align*}

Observe: $\sf G_1 \ge G_2.$

$\sf G_1 - G_2 = q+q^4+q^5+q^6+q^7+2q^8+2q^9+\cdots.$

Let $\sf G_3 = (G_1 - G_2)/q = 1+q^3+q^4+q^5+q^6+2q^7+2q^8+\cdots.$

Let $\sf G_4 = (G_2 - G_3)/q^2 = 1+q^4+q^5+q^6+2q^7+2q^8+\cdots$.

Continue:
Let $\sf \bbox{G_i = (G_{i-2} - G_{i-1})/q^{i-2}}.$

Observe that each $\sf G_i = 1 + q^i + \cdots$ : Empirical Hypothesis

Proof of the Empirical Hypothesis not only answers Ehrenpreis's question but also immediately leads to a proof of RR.

## 3.2 Some remarks

• A totally combinatorial proof.
• A "pure $\sf q$" proof. Unlike Andrews' "Chapter 7" "$\sf (x,q)$"-proof. The "$\sf x$" in that proof is yet to be interpreted in the $\sf Z$-algebraic setup.
• Has the same "philosophical" starting point as Lepowsky-Wilson: Arriving at the sum sides, given the product sides.

$\sf\prod \longrightarrow \sum$

• Treats both the RR identities at once
• Lepowsky-Wilson's $\sf Z$-algebraic approach treats the identities one-by-one, i.e., they consider one module at a time.
• Motivated proof would correspond to going back and forth between modules.

## 3.3 Generalizations

• Lepowsky-Zhu generalized the above proof to the full family of Gordon-Andrews (odd modulus) identities.
• The "shelf" picture, implicit in Andrews-Baxter's motivated proof, became transparent.
• For a fixed $\sf k$, the zeroth shelf consists of the given $\sf k$ products.
• Each successive shelf is obtained recursively from the previous shelf.
• One moves up a shelf by subtracting the formal power series on the current shelf and dividing the result by a pure power of $\sf q.$
• Coulson-Kanade-Lepowsky-McRae-Qi-Russell-Sadowski gave a motivated proof of the somewhat analogous Goellnitz-Gordon-Andrews identities.
• Replacing $\sf x$ with successively higher powers of $\sf q$ in Andrews's "Chapter 7" $\sf(x,q)$-proof yields the motivated proof of these identities. (Appendices A and C of [CKLMQRS]).
• The Empircal Hypothesis in these works is therefore not Empirical, since it is known in advance "what is going to happen".

• A starkly different case of Andrews-Bressoud (even modulus) identities is handled in Kanade-Lepowsky-Russell-Sills. New phenomena emerge!

# 4 The bridge

## 4.1 "Categorification"

• Interpreting various aspects of patition identities using "intertwining operators" --- A broad idea of J. Lepowsky.
• Capparelli-Lepowsky-Milas, Calinescu, Calinescu-Lepowsky-Milas, Sadowski have "categorified" Rogers-Ramanujan, Rogers-Selberg, Euler and other recursions using other vertex-algebraic settings.
• Interpret the steps in the a motivated proof using exact sequences of standard modules with the maps coming from "twisted relativized intertwining operators" --- an idea of J. Lepowsky and A. Milas.

$\sf G_3 = (G_1 - G_2)/q \quad$ corresponds to the exact sequence $\quad\sf 0 \longrightarrow \Lambda_{G_2} \longrightarrow \Lambda_{G_1} \longrightarrow ? \longrightarrow 0.$

• Will provide a useful insight into the representation theory of vertex algebras.
• Will facilitate the discovery of further such families of identities.
• Still not complete, only a relevant level-1 "(twisted) abelian intertwining algebra" structure is known.

# 5 Ghost Series

## 5.1 Motivated proof of Andrews-Bressoud identities

• Since both Gordon-Andrews and Andrews-Bressoud deal with $\sf A_1^{(1)}$-modules,
and since $\sf Z$-algebraic approach treats both families on a more-or-less equal footing,
motivated proofs of both the identities are important.
• Due to the additional "parity condition" in the sum sides of Andrews-Bressoud identities, the natural recursions for obtaining successive shelves involve division by more complicated expressions such as a sum of two pure powers of $\sf q$.

$\sf B_{(k-1)(j+1)+i}= \dfrac{B_{(k-1)j+k-i+1}-B_{(k-1)j+k-i+3}}{q^{(j+1)(i-2)}(1+q^{j+1})}$

• Such a division is difficult to "categorify" and is not motivated either.

## 5.2 The Ghosts

• Introduce "Ghost Series" which facilitate the passage to successive shelves via division by pure powers of $\sf q$ (Kanade-Lepowsky-Russell-Sills).
• Key idea: Given the $\sf j^\text{th}$ shelf of official series, introduce relations which simultaneously define the ghosts on the $\sf j^\text{th}$-shelf and also provide a "motivated" passage to $\sf (j+1)^\text{th}$ shelf. These relations have the required shape, in that, involve divisions by pure powers of $\sf q.$

$\sf B_{(k-1)(j+1)+i} = \dfrac{B_{(k-1)j + k-i+1 } - \mathfrak{B}_{ (k-1)j +k-i+2 } } { q^{ (j+1) (i-1) } } = \dfrac{\mathfrak{B}_{ (k-1)j + k-i+2 } - B_{ (k-1)j+k-i+3 } }{ q^{ (j+1)( i - 2) } }$

• In this way, obtain the Empirical Hypothesis which yeilds the Andrews-Bressoud identities.
• Once the main theorem is proved, one also immediately gets combinatorial interpretations of the ghost series. They differ from the official series only in the parity condition.
• The ghosts form a new family of partition identities.
• Unlike the official series on the zeroth shelf (which are products), the ghosts on the zeroth shelf don't click into elegant product forms.
• The ghost series and the relations just mentioned were discovered purely empirically, and thus the Empirical Hypothesis is indeed "empirical!"

## 5.3 Further questions

• Where do the ghosts recide in a vertex algebraic setting?
• Modularity properties?
• Representations as multisums?
• What other identities can be handled with the help of ghosts?

 "I've caught a cold," the Thing replies, "Out there upon the landing." I turned to look in some surprise, And there, before my very eyes, A little Ghost was standing!    He trembled when he caught my eye, And got behind a chair. "How came you here," I said, "and why? I never saw a thing so shy. Come out! Don't shiver there!"  Phantasmagoria, Lewis Carroll