Classification of Multiplicity-Free Kronecker Products

Classiﬁcation of Multiplicity-Free Kronecker Products

Christine Bessenrodt (Leibniz University Hannover) Jeffrey Remmel (University of California, San Diego) Stephanie van Willigenburg (University of British Columbia) and Sami Assaf (University of Southern California) Christopher Bowman (City University London) Angela Hicks (Stanford University) Vasu Tewari (University of British Columbia) May 10–May 17 2015

1 Overview of the Field

Schur functions have been a central area of research dating from the time of Cauchy in 1815 (bialternants [7]) to the present day work of Field’s medallists such as Tao (the hive model [8]) and Okounkov (Schur positivity [9]). This is due to their ubiquitous nature in mathematics and beyond. Schur functions are labelled by weakly decreasing sequences of positive integers, called partitions. Given two Schur functions, they can be ‘multiplied’ in three different ways: outer product, inner product, and plethysm. We wish to understand the coefficients that arise in expanding such a product with respect to the basis of Schur functions. The outer product is the most well understood, the coefficients arising are the Littlewood–Richardson coefficients and there is an efficient combinatorial description of these coefficients, called the Littlewood– Richardson rule. In the case of the inner and plethysm products, the coefficients are called the Kronecker and plethysm coefficients respectively, and they are substantially more difficult to understand. The determi- nation of the classical Clebsch-Gordan coefficients arising in the decomposition of tensor products amounts in the case of the symmetric groups to a search for an efficient combinatorial description of the Kronecker coefficients; it is one of the central problems of algebraic combinatorics today. Additionally, outside of pure mathematics, this Kronecker problem has gained great traction of late due to its deep connections with quantum information theory and the central role it plays within Geometric Com- plexity Theory, an approach that seeks to settle the celebrated P versus NP problem via Geometric Complexity Theory. The P versus NP problem is one of the several $1,000,000 Millennium Prize Problems set by the Clay Mathematics Institute. Geometric Complexity Theory poses several conjectures on the computation of Kronecker coefficients, even for special partitions, and on deciding whether a given coefficient is non-zero. It is conjectured that the decidability problem is in ]P (is ‘easy’), whereas calculation of a given coefficient is in ]NP (is ‘hard’). It is known that the problem of computing Kronecker coefficients is ]P -hard [6].

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2 Recent Developments and Open Problems

The search for an efficient combinatorial description of the Kronecker coefficients (of a similar flavour as the Littlewood-Richardson rule) has formed a very active research area for several decades. Along the way, many beautiful formulas have been discovered for specific cases, including • products indexed by a pair of hooks or two-part partitions [10], [11], [12], [13], and [5]; • constituents indexed by partitions of small depth [16], by partitions of small Durfee size [1], or by hooks [4]; • products with few homogenous components [2] and [3]; • products indexed by partitions of near-rectangular shape [15]. These formulas serve both as inspiration, and as a reminder that there is still a great amount of work to do. One further avenue, which has yet to be pursued, is to classify Kronecker products whose coefficients satisfy certain bounds. In this direction, in 1999 Bessenrodt conjectured a complete classification of multiplicity- free Kronecker products of Schur functions (or equivalently, of irreducible characters of the symmetric groups), that is, pairs of Schur functions whose Kronecker product only yields coefficients equal to 0 or 1. During the recent American Institute of Mathematics workshop Combinatorics and complexity of Kro- necker coefficients our group made progress towards Bessenrodt’s conjecture for the case where one Schur function was indexed by a hook, and the other Schur function was indexed by an arbitrary partition. Therefore the main aim for our Focused Research Group was to completely resolve Bessenrodt’s 1999 conjectured classification of multiplicity-free Kronecker products.

3 Scientiﬁc Progress Made

Our Focused Research Group was completely successful in proving the following theorem, which had been a conjecture of Bessenrodt, dating from 1999; at that time, it was known that all the products on the list were multiplicity-free and the problem remained to show that the list was complete.

Theorem 3.1. Let λ, µ be partitions of n ∈ N. Then the Kronecker product sλ ⊗ sµ of the Schur functions sλ, sµ is multiplicity-free if and only if the partitions λ, µ satisfy one of the following (up to conjugation of one or both of the partitions): 1. One of the partitions is (n), and the other one is arbitrary; 2. one of the partitions is (n − 1, 1), and the other one is a fat hook; 3. n = 2k + 1 and λ = (k + 1, k) = µ, or n = 2k and λ = (k, k) = µ; 4. n = 2k, one of the partitions is (k, k), and the other one is one of (k + 1, k − 1), (n − 3, 3) or a hook; 5. one of the partitions is a rectangle, and the other one is one of (n − 2, 2), (n − 2, 12); 6. the partition pair is one of the pairs ((33), (6, 3)), ((33), (5, 4)), and ((43), (62)). It is worth pointing out that a classification of multiplicity-free outer products of Schur functions was obtained by Stembridge [14]; the application of his result (and a generalization due to Gutschwager) plays an important role in the proof of Theorem 3.1. A key facet of our proof was to deal first with some special situations which are close to the products on the list and which arise as critical situations in the final induction proof. Crucial aspects are a detailed case analysis and a delicate application of both a recursion formula for Kronecker products due to Dvir and a semigroup property of Kronecker coefficients observed by Manivel; for small size, also computer calculations have been used. Towards the classification result we showed first:

• Let λ be a hook or a 2-part partition. Then the Kronecker product sλ ⊗ sµ is multiplicity-free if and only if it appears on the list given by Theorem 3.1. 3

As part of the induction argument in the proof of Theorem 3.1, we next dealt with products sλ ⊗sµ where λ was assumed to be a rectangle, and then more generally with the case of a fat hook λ. A further important feature of the inductive approach to the classification result stated in Theorem 3.1 was to employ its close connection to a more general classification conjecture on multiplicity-free products of skew Schur functions (formulated by Bessenrodt some years ago). With these results and insights as our foundation, we were able to extend our case analysis and use our techniques once more to eventually prove Theorem 3.1. As just indicated, this then also led to the confirma- tion of the more general classification conjecture for multiplicity-free products of skew Schur functions.

4 Outcome of the Meeting

Further to completing our project, from which a journal article will result, we were also able to become fully versed in the strengths and weaknesses of powerful techniques such as Dvir recursion and Manivel’s semigroup property, and learn new techniques related to quasisymmetric functions. These were facilitated by informal lectures given by Bessenrodt and Remmel, respectively. We would like to thank BIRS for this indispensable Focused Research Group opportunity, which was crucial to the success of our project.

References

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[5] A. Brown, S. van Willigenburg and M. Zabrocki, Expressions for Catalan Kronecker products, Paciﬁc J. Math. 248 (2010), 31–48. [6] P. Burgisser¨ and C. Ikenmeyer, The complexity of computing Kronecker coefﬁcients, DMTCS proc. AJ (2008), 357–368.

[7] A. Cauchy, Memoire´ sur les fonctions qui ne peuvent obtenir que deux valeurs egales´ et de signes con- traires par suite des transpositions oper´ ees´ entre les variables qu’elles renferment, Journal de l’Ecole´ Polytechnique xviie Cahier, Tome x (1815), 91–169.

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[11] J. Remmel, Formulas for the expansion of the Kronecker products S(m, n)⊗S(1p−r, r) and S(1k, 2l)⊗ S(1p−r, r), Discrete Math. 99 (1992), 265–287. [12] J. Remmel and T. Whitehead, On the Kronecker product of Schur functions of two row shapes, Bull. Belg. Math. Soc. 1 (1994), 649–683. 4

[13] M. Rosas, The Kronecker product of Schur functions indexed by two-row shapes or hook shapes, J. Algebraic Combin. 14 (2001), 153–173. [14] J. Stembridge, On multiplicity-free products of Schur functions, Ann. Comb. 5 (2001), 113–121. [15] V. Tewari, Kronecker coefﬁcients for some near-rectangular partitions, http://arxiv.org/abs/1403.5327. [16] E. Vallejo, On the Kronecker product of the irreducible characters of the symmetric group, Pub. Prel. Inst. Mat. UNAM 526 (1997).