The Algebraic Combinatorics Seminar

Schedule

In reverse-chronological order.

Date Speaker Title (click titles for abstract)

Mar 24, 2017 Mike Zabrocki Crystals and multiset tableaux.

Mar 17, 2017 Farid and Shu Xiao New Theta developments.

Mar 10, 2017 Mike Zabrocki Plethysm, crystals and multiset tableaux.
We will see how the multi-set tableaux arise as an example of a coefficient of plethysm and (partially) explain why we were able to use the crystal graph structure of tensors of one row tableaux to interpret them as multi-set tableaux. We provide a conjectured combinatorial interpretation for the coefficient of ${\tilde s}_\lambda$ in $s_{(a,1^b)}$ and use plethysm and crystals to try to extend this conjecture to the coefficient of ${\tilde s}_\lambda$ in $s_\mu$.

Mar 3, 2017 Farid and Shu Xiao New Theta developments.

Feb 24, 2017 Reading Week

Feb 17, 2017 Carolina Benedetti and Nantel Bergeron Quantum Murnaghan-Nakayama rule.

Feb 10, 2017 Carolina Benedetti and Nantel Bergeron Any quantum forest can be cyclicly shifted to a classical interval

Feb 3, 2017 Mike Zabrocki Crystals and multiset tableaux.
If we think of a multiset as a sequence of increasing entries then one row tableaux are in bijection with multi-sets. We can use this idea to interpret multiset tableaux as tensors of crystals of one row shapes. The idea works well and provides a combinatorial interpretation for the ${\tilde s}_\lambda$ expansion of $s_\mu$ for small values of $\mu$.

Jan 27, 2017 No talk

Jan 20, 2017 Franco Saliola and Carolina Benedetti interval in the quantum "order", cyclic class and existence of a classical interval in some cyclic class. [click to reveal PDF of slides]
PDF of Slides

13 Jan. 2017 Farid Aliniaeifard The theta map

(1) The start is the theta map for Sym (Symmetric functions) in Macdonald. Read Chap 3 Section 8 (its pretty much self contain if you read the first chapter). Look at Exercise 10, the map is there... but called $\varphi$.
(2) The map was extended by Stembridge to QSym. In his case he extended the theory of Stanley's P-partition to understand Q-Schur functions:
Enriched P-partitions. Trans. Amer. Math. Soc. 349 (1997), no. 2, 763-788.
See remark 3.2 in the above paper
(3) We [Steph-Stef-Frank-Nantel] show that this is a Hopf map and is related to Eulerian poset
Discrete Mathematics, Vol. 246, 1-3 (2002) 57-66.
JCT-A, Vol. 91, 1-2 (2000) 84-110.
(4) Now in this context, the peak algebra takes its full importance as it encode the generalized Dehn-Sommerville relations. The paper [Marcelo-Frank-Nantel] fully show the context and the importance of these algebra maps:
Compositio Mathematica, Vol. 142, 1 (2006) 1-30.
See example 4.9
(5) Now, to push this further, you may look at [Marcelo-Kathreen-Nantel]:
Trans. Amer. Math. Soc., Vol. 356, 7 (2004) 2781-2824.
Section 7.4 contains some better understanding of the tetha map and some extension to type B.
(6) These are the kind of map we failed to lift to the Malvenuto-Reutenauer (MR) algebra.
(a) The odd subalgebra of MR is NOT the homomorphic image of a morphism (as far as I know)
(b) As I remember (with Mike) we programed, "what are the Hopf morphism of MR that would commute with the other maps involves". If my memory serve me well the answer was "none".
So we now replace MR by V (Vargas. I am sure it is elsewhere too, but cannot find it):
http://www.dmtcs.org/pdfpapers/dmAT0172.pdf

16 Dec. 2016 Franco Saliola and Carolina Benedetti Conjecture for hooks: Quantum implies classical?
It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.
By now... I hope we should have a proof... :)

9 dec. 2016 Shu Xiao Li Find the combinatorial rule for the branching rule between GL(n) and the symmetric group
See the abstract on Sept 9 for a description of the problem, references and programs. The presenters will lead the discussion about their progress in finding an injection of the multi-set tableaux of content $\mu$ to the multi-set tableaux of content $s_i(\mu)$.

2 Dec. 2016 Carolina Benedetti Conjecture for hooks: Quantum implies classical?
It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.
We discussed progress in understanding proof of Theorem 8 in MPP.
We also discussed some operators on multi-set tableaux that use ideas in the following two references.
[1] K. O'hara, Unimodality of Gaussian Coefficients: A Constructive Proof, J. Comb. Theory, Series A, 53, (1990), pp. 29--52.
[2] J. Remmel, M. Shimozono, A simple proof of the Littlewood-Richardson rule and applications, Disc. Math., vol 193, Issues 1--3 (1998), pp. 257--266.

25 Nov. 2016 Shu Xiao Li and John Campbell Find the combinatorial rule for the branching rule between GL(n) and the symmetric group
See the abstract on Sept 9 for a description of the problem, references and programs. The presenters will lead the discussion about their progress in finding an injection of the multi-set tableaux of content $\mu$ to the multi-set tableaux of content $s_i(\mu)$.

18 Nov. 2016 Franco Saliola Conjecture for hooks: Quantum implies classical?
It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.

11 Nov. 2016 Mike Zabrocki Find the combinatorial rule for the branching rule between GL(n) and the symmetric group
See Sept 9 abstract for more information and suggested reading.

4 Nov. 2016 Frank Sottile and Nantel Bergeron Conjecture for hooks: Quantum implies classical?
It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.

28 Oct. 2016 Nantel Bergeron computations in quantum Fomin-Kirrilov algebra.
It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.

21 Oct. 2016 Carolina Benedetti and Laura Colmenarejo Discussion on the quantum Bruhat network
As a first step, we study the paper SCHUR TIMES SCHUBERT VIA THE FOMIN-KIRILLOV ALGEBRA in order to understand better the quatum pieri rule for Schubert calculus.

14 Oct. 2016 Carolina Benedetti Discussion on the quantum Bruhat network
Nantel plan to
- Justify the definition of power sum in quantum case
- pose the problem of understanding Quantum pier rule in term of increasing path in a quantum k-Bruhat orderâ
- sketch a second proof of Frank's result in term of paths only

7 Oct. 2016 NO SEMINAR We go to Convexity workshop

30 Sep. 2016 Frank Sottile A Murnaghan-Nakayama rule for Schubert polynomials in quantum cohomology

23 Sep. 2016 Alex Woo List of problems related to Reduced Total order
We have a nice list of problems related to the order R and the order weak(R)
-1- Find an injection from $Inv(w)$ to $Exc(w)\times Def(w)$
-2- Show that $L_R(w) = L_R( \gamma^{-1} w \gamma )$
-3- Show that $L_R(w) = \sum_{(a,b)\in Inv(w)} \chi_{w}(a,b)$ where $\chi_{w}(a,b)$ is zero when $a$ or $b$ are fixed point of $w$, is 1 if $a\lt w(a)$ and $b\gt w(b)$ and is -1 otherwise. The right hand side of this formula comes from the rank function in the [Bergeron-Sottile] (Grassmannian) order. see Section 3.2 of
Duke mathematical journal, Vol. 95-2 (1998) 373-423
-4- Show that the order of [Bergeron-Sottile] is the same as the weak(R) order.
This is shown by Bennett and Blok in Partial orders generalizing the weak order on Coxeter groups, JCT-A Vol. 102 (2003) 331-346
-5- Show that the maximal element of the order R are exactly the non-trivial power of the long cycle $(1, 2, ... , n)$.
-6- Can we say something about the interval $[Id,w]_R$ where $w$ is a maximal element of the R-order. (remark that for the same max element the interval $[I,w]_{weak(R)}$ are very nice, they should be the Young lattice in the rectagle defined by $w$.)

A simple SAGE program to compute the rank and Hilbert function of this order follow this link.

16 Sep. 2016 Alex Woo Understanding the "reduced absolute" order of the symmetric group
The reference for reduced reflection length is p. 26 of Bagno-Biagioli-Novick-me:
http://arxiv.org/pdf/1507.01180.pdf
The only reasonable way I know to calculate it (for S_n) is to combine Proposition 6.6 (which is an equality for S_n) and Theorem 1.1 in Petersen-Tenner:
http://arxiv.org/pdf/1202.4765.pdf
As far as I know, the notion of "reduced absolute order" is not defined in print anywhere.
Some of the kinds of questions one might be interested in were answered for absolute order (the order based on reflection length) in
http://arxiv.org/pdf/0706.1405.pdf
and various papers it refers to. (For Bruhat order (the order based on length), these have been well known for a very long time.)
[Added by Nantel: It seems like it is very interesting to look at the "nil"-operators algebra of these poset as in Benedetti-Bergeron: http://arxiv.org/pdf/1307.2528.pdf
my initial computation reveals that it is very interesting in these cases]

9 Sep. 2016 Nantel Bergeron Four Suggested problem for this year (Mike Zabrocki, Frank Sottile, Alex Woo and Nantel Bergeron).
Mike: Find the combinatorial rule for the branching rule between GL(n) and the symmetric group click here to see more
Take an irreducible $Gl(n)$ representation and consider the subgroup of permutation matrices contained in the group of all invertible matrices. The goal of this problem will be to determine how this irreducible decomposes into permutation group irreducible representations. Representation theory gives one answer to this question, namely, find the Schur expansion of the Frobenius image of the character of the irreducible $Gl(n)$ representation. Since the irreducible $Gl(n)$ represenation has character equal to a Schur function $s_\lambda[X_n]$, we are asking for the Schur expansion of $$\sum_{\mu \vdash n} s_\lambda[\Xi_\mu] \frac{p_\mu}{z_\mu}$$ where $f[\Xi_\mu]$ means evaluate the symmetric function $f$ at the eigenvalues of the permutation matrix with cycle structure $\mu$.

A new approach to this problem is to use a tool of a basis of the symmetric functions which represents the irreducible characters of the symmetric group. The problem of finding the coefficient of $s_{(n-|\nu|,\nu)}$ in the equation above is equivalent (for $n>|\nu|+\nu_1$) to asking for the coefficient of ${\tilde s}_\nu$ in $s_\lambda$ where ${\tilde s}_\nu$ is the irreducible character basis of the symmetric group represented as permutation matrices (see reference [2]). We don't have a combinatorial interpretation for the transition between the Schur and the irreducible character basis, but we do between the homogeneous (complete) basis and the irreducible character basis and between the homogeneous and the Schur basis.

Essentially, we are trying to complete the following diagram.

The coefficient of $s_\lambda$ in $h_\mu$ is equal to the number of column strict tableaux of shape $\lambda$ and content $\mu$. The coefficient of ${\tilde s}_\lambda$ in $h_\mu$ is equal to the number of column strict multi-set valued tableaux of skew shape $(k,\lambda)/(\lambda_1)$ and content $\mu$. The column strictness of the multiset tableau is with respect to lexicographic order on the multi-sets...but technically you can choose any total order on multisets if it makes the problem easier to solve.

If you would like to compute data for this problem, some programs are already implemented in Sage.
sage: Sym = SymmetricFunctions(QQ) sage: s = Sym.schur() sage: st = Sym.irreducible_symmetric_group_character() sage: st(s[2,1]) st[] + 3*st[1] + 2*st[1, 1] + 2*st[2] + st[2, 1] sage: s[2,1].character_to_frobenius_image(5) s[2, 2, 1] + 2*s[3, 1, 1] + 2*s[3, 2] + 3*s[4, 1] + s[5]

Sage users can execute the following two commands to get classes MultisetPartition(s) and MultisetTableau(x) that may be helpful for experimentation.

load('http://garsia.math.yorku.ca/~zabrocki/multisetpartition.py') load('http://garsia.math.yorku.ca/~zabrocki/multisettableau.py')

[1] K. Nishiyama, an unpublished paper about the restriction problem from $Gl(n)$ to $S_n$ and its connection to plethysm.
Restriction of the irreducible representations of $ GL_n $ to the symmetric group ${S}_n$
[2] R. Orellana and M. Zabrocki, Symmetric Group Characters as Symmetric Functions
[3] R.C. King, Branching rules for $GL(N) \supset \Sigma_m$ and the evaluation of inner plethysm
[4] This seminar from 2006-2007 and again in 2009-2010 and notes developed from that discussion.
[5] T. Scharf, J. Y. Thibon, A Hopf-algebra approach to inner plethysm, Adv. in Math. 104 (1994), pp. 30--58.
[6] D.E. Littlewood, Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric Groups, Canad. J. Math., {\bf 10}, 1958, 17--32.

Frank: Prove conjectured Murnaghan-Nakayama for quantum Schubert polynomials click here to see more
[1] A. Morrison, A Murnaghan-Nakayama rule for Schubert polynomials, FPSAC 2014
[2] A. Morrison, F. Sottile, Proof for ordinary cohomology, Annals of Combinatorics
[3] K. Meszaros, G. Panova, A. Postnikov, Schur times Schubert via the Fomin-Kirillov algebra, Electronic J. Combinatorics, (2013).
[4] A. Postnikov, On a quantum version of Pieri's formula, Advances in geometry, Progr. Math., vol. 172, Birkhauser Boston, Boston, MA, 1999, pp. 371--383.

Alex: Understanding the "Reduce" order of the symmetric group
Nantel: Complete the square for the Theta map.

Year	Topic
2015-2016	Positroid and Matroid/ Hopf algebras and quotients.
2014-2015	Fiboland, Symmetric and non symmetric functions
2013-2014	Fiboland, a world of Catalan and Fibonacci numbers
2012-2013	NSym and the Immaculate Basis
2011-2012	k-Schur functions and affine permutations
2010-2011	Littlewood Richardson rule k-Schur functions.
2009-2010	Idempotents and weakly ordered semigroups. (q,t) Catalan Numbers.
2008-2009	Littlewood-Richardson Rule, Shifted Tableaux and P-Schur functions
2007-2008	Open problems around k-Schur functions and non-commutative symmetric functions
2006-2007	Open problems
2005-2006	Cluster Algebras and Quivers
Spring 2005	Formal languages and analytic classes of functions
Fall 2004	(Quasi-) Symmetric functions in noncommutative variables and applications
Winter 2003	Crystal Bases and Representation Theory, Super-algebras, etc.
Fall 2003	Quasi-Symmetric functions and applications
Fall 2002	Crystal Bases and Representation Theory

Date	Speaker	Title (click titles for abstract)
Mar 24, 2017	Mike Zabrocki	Crystals and multiset tableaux.
Mar 17, 2017	Farid and Shu Xiao	New Theta developments.
Mar 10, 2017	Mike Zabrocki	Plethysm, crystals and multiset tableaux. We will see how the multi-set tableaux arise as an example of a coefficient of plethysm and (partially) explain why we were able to use the crystal graph structure of tensors of one row tableaux to interpret them as multi-set tableaux. We provide a conjectured combinatorial interpretation for the coefficient of ${\tilde s}_\lambda$ in $s_{(a,1^b)}$ and use plethysm and crystals to try to extend this conjecture to the coefficient of ${\tilde s}_\lambda$ in $s_\mu$.
Mar 3, 2017	Farid and Shu Xiao	New Theta developments.
Feb 24, 2017	Reading Week
Feb 17, 2017	Carolina Benedetti and Nantel Bergeron	Quantum Murnaghan-Nakayama rule.
Feb 10, 2017	Carolina Benedetti and Nantel Bergeron	Any quantum forest can be cyclicly shifted to a classical interval
Feb 3, 2017	Mike Zabrocki	Crystals and multiset tableaux. If we think of a multiset as a sequence of increasing entries then one row tableaux are in bijection with multi-sets. We can use this idea to interpret multiset tableaux as tensors of crystals of one row shapes. The idea works well and provides a combinatorial interpretation for the ${\tilde s}_\lambda$ expansion of $s_\mu$ for small values of $\mu$.
Jan 27, 2017	No talk
Jan 20, 2017	Franco Saliola and Carolina Benedetti	interval in the quantum "order", cyclic class and existence of a classical interval in some cyclic class. [click to reveal PDF of slides] PDF of Slides
13 Jan. 2017	Farid Aliniaeifard	The theta map (1) The start is the theta map for Sym (Symmetric functions) in Macdonald. Read Chap 3 Section 8 (its pretty much self contain if you read the first chapter). Look at Exercise 10, the map is there... but called $\varphi$. (2) The map was extended by Stembridge to QSym. In his case he extended the theory of Stanley's P-partition to understand Q-Schur functions: Enriched P-partitions. Trans. Amer. Math. Soc. 349 (1997), no. 2, 763-788. See remark 3.2 in the above paper (3) We [Steph-Stef-Frank-Nantel] show that this is a Hopf map and is related to Eulerian poset Discrete Mathematics, Vol. 246, 1-3 (2002) 57-66. JCT-A, Vol. 91, 1-2 (2000) 84-110. (4) Now in this context, the peak algebra takes its full importance as it encode the generalized Dehn-Sommerville relations. The paper [Marcelo-Frank-Nantel] fully show the context and the importance of these algebra maps: Compositio Mathematica, Vol. 142, 1 (2006) 1-30. See example 4.9 (5) Now, to push this further, you may look at [Marcelo-Kathreen-Nantel]: Trans. Amer. Math. Soc., Vol. 356, 7 (2004) 2781-2824. Section 7.4 contains some better understanding of the tetha map and some extension to type B. (6) These are the kind of map we failed to lift to the Malvenuto-Reutenauer (MR) algebra. (a) The odd subalgebra of MR is NOT the homomorphic image of a morphism (as far as I know) (b) As I remember (with Mike) we programed, "what are the Hopf morphism of MR that would commute with the other maps involves". If my memory serve me well the answer was "none". So we now replace MR by V (Vargas. I am sure it is elsewhere too, but cannot find it): http://www.dmtcs.org/pdfpapers/dmAT0172.pdf
16 Dec. 2016	Franco Saliola and Carolina Benedetti	Conjecture for hooks: Quantum implies classical? It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule. By now... I hope we should have a proof... :)
9 dec. 2016	Shu Xiao Li	Find the combinatorial rule for the branching rule between GL(n) and the symmetric group See the abstract on Sept 9 for a description of the problem, references and programs. The presenters will lead the discussion about their progress in finding an injection of the multi-set tableaux of content $\mu$ to the multi-set tableaux of content $s_i(\mu)$.
2 Dec. 2016	Carolina Benedetti	Conjecture for hooks: Quantum implies classical? It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule. We discussed progress in understanding proof of Theorem 8 in MPP. We also discussed some operators on multi-set tableaux that use ideas in the following two references. [1] K. O'hara, Unimodality of Gaussian Coefficients: A Constructive Proof, J. Comb. Theory, Series A, 53, (1990), pp. 29--52. [2] J. Remmel, M. Shimozono, A simple proof of the Littlewood-Richardson rule and applications, Disc. Math., vol 193, Issues 1--3 (1998), pp. 257--266.
25 Nov. 2016	Shu Xiao Li and John Campbell	Find the combinatorial rule for the branching rule between GL(n) and the symmetric group See the abstract on Sept 9 for a description of the problem, references and programs. The presenters will lead the discussion about their progress in finding an injection of the multi-set tableaux of content $\mu$ to the multi-set tableaux of content $s_i(\mu)$.
18 Nov. 2016	Franco Saliola	Conjecture for hooks: Quantum implies classical? It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.
11 Nov. 2016	Mike Zabrocki	Find the combinatorial rule for the branching rule between GL(n) and the symmetric group See Sept 9 abstract for more information and suggested reading.
4 Nov. 2016	Frank Sottile and Nantel Bergeron	Conjecture for hooks: Quantum implies classical? It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.
28 Oct. 2016	Nantel Bergeron	computations in quantum Fomin-Kirrilov algebra. It seems that for any (non-zero) operator apearing in the MPP formula for hooks, we can find a permutation such that the classical action of this term is non-zero. This would lead to a proof of the Quantum M-N rule.
21 Oct. 2016	Carolina Benedetti and Laura Colmenarejo	Discussion on the quantum Bruhat network As a first step, we study the paper SCHUR TIMES SCHUBERT VIA THE FOMIN-KIRILLOV ALGEBRA in order to understand better the quatum pieri rule for Schubert calculus.
14 Oct. 2016	Carolina Benedetti	Discussion on the quantum Bruhat network Nantel plan to - Justify the definition of power sum in quantum case - pose the problem of understanding Quantum pier rule in term of increasing path in a quantum k-Bruhat orderâ - sketch a second proof of Frank's result in term of paths only
7 Oct. 2016	NO SEMINAR	We go to Convexity workshop
30 Sep. 2016	Frank Sottile	A Murnaghan-Nakayama rule for Schubert polynomials in quantum cohomology
23 Sep. 2016	Alex Woo	List of problems related to Reduced Total order We have a nice list of problems related to the order R and the order weak(R) -1- Find an injection from $Inv(w)$ to $Exc(w)\times Def(w)$ -2- Show that $L_R(w) = L_R( \gamma^{-1} w \gamma )$ -3- Show that $L_R(w) = \sum_{(a,b)\in Inv(w)} \chi_{w}(a,b)$ where $\chi_{w}(a,b)$ is zero when $a$ or $b$ are fixed point of $w$, is 1 if $a\lt w(a)$ and $b\gt w(b)$ and is -1 otherwise. The right hand side of this formula comes from the rank function in the [Bergeron-Sottile] (Grassmannian) order. see Section 3.2 of Duke mathematical journal, Vol. 95-2 (1998) 373-423 -4- Show that the order of [Bergeron-Sottile] is the same as the weak(R) order. This is shown by Bennett and Blok in Partial orders generalizing the weak order on Coxeter groups, JCT-A Vol. 102 (2003) 331-346 -5- Show that the maximal element of the order R are exactly the non-trivial power of the long cycle $(1, 2, ... , n)$. -6- Can we say something about the interval $[Id,w]_R$ where $w$ is a maximal element of the R-order. (remark that for the same max element the interval $[I,w]_{weak(R)}$ are very nice, they should be the Young lattice in the rectagle defined by $w$.) A simple SAGE program to compute the rank and Hilbert function of this order follow this link.
16 Sep. 2016	Alex Woo	Understanding the "reduced absolute" order of the symmetric group The reference for reduced reflection length is p. 26 of Bagno-Biagioli-Novick-me: http://arxiv.org/pdf/1507.01180.pdf The only reasonable way I know to calculate it (for S_n) is to combine Proposition 6.6 (which is an equality for S_n) and Theorem 1.1 in Petersen-Tenner: http://arxiv.org/pdf/1202.4765.pdf As far as I know, the notion of "reduced absolute order" is not defined in print anywhere. Some of the kinds of questions one might be interested in were answered for absolute order (the order based on reflection length) in http://arxiv.org/pdf/0706.1405.pdf and various papers it refers to. (For Bruhat order (the order based on length), these have been well known for a very long time.) [Added by Nantel: It seems like it is very interesting to look at the "nil"-operators algebra of these poset as in Benedetti-Bergeron: http://arxiv.org/pdf/1307.2528.pdf my initial computation reveals that it is very interesting in these cases]
9 Sep. 2016	Nantel Bergeron	Four Suggested problem for this year (Mike Zabrocki, Frank Sottile, Alex Woo and Nantel Bergeron). Mike: Find the combinatorial rule for the branching rule between GL(n) and the symmetric group click here to see more Take an irreducible $Gl(n)$ representation and consider the subgroup of permutation matrices contained in the group of all invertible matrices. The goal of this problem will be to determine how this irreducible decomposes into permutation group irreducible representations. Representation theory gives one answer to this question, namely, find the Schur expansion of the Frobenius image of the character of the irreducible $Gl(n)$ representation. Since the irreducible $Gl(n)$ represenation has character equal to a Schur function $s_\lambda[X_n]$, we are asking for the Schur expansion of $$\sum_{\mu \vdash n} s_\lambda[\Xi_\mu] \frac{p_\mu}{z_\mu}$$ where $f[\Xi_\mu]$ means evaluate the symmetric function $f$ at the eigenvalues of the permutation matrix with cycle structure $\mu$. A new approach to this problem is to use a tool of a basis of the symmetric functions which represents the irreducible characters of the symmetric group. The problem of finding the coefficient of $s_{(n-\|\nu\|,\nu)}$ in the equation above is equivalent (for $n>\|\nu\|+\nu_1$) to asking for the coefficient of ${\tilde s}_\nu$ in $s_\lambda$ where ${\tilde s}_\nu$ is the irreducible character basis of the symmetric group represented as permutation matrices (see reference [2]). We don't have a combinatorial interpretation for the transition between the Schur and the irreducible character basis, but we do between the homogeneous (complete) basis and the irreducible character basis and between the homogeneous and the Schur basis. Essentially, we are trying to complete the following diagram. The coefficient of $s_\lambda$ in $h_\mu$ is equal to the number of column strict tableaux of shape $\lambda$ and content $\mu$. The coefficient of ${\tilde s}_\lambda$ in $h_\mu$ is equal to the number of column strict multi-set valued tableaux of skew shape $(k,\lambda)/(\lambda_1)$ and content $\mu$. The column strictness of the multiset tableau is with respect to lexicographic order on the multi-sets...but technically you can choose any total order on multisets if it makes the problem easier to solve. If you would like to compute data for this problem, some programs are already implemented in Sage. `sage: Sym = SymmetricFunctions(QQ) sage: s = Sym.schur() sage: st = Sym.irreducible_symmetric_group_character() sage: st(s[2,1]) st[] + 3st[1] + 2st[1, 1] + 2st[2] + st[2, 1] sage: s[2,1].character_to_frobenius_image(5) s[2, 2, 1] + 2s[3, 1, 1] + 2s[3, 2] + 3s[4, 1] + s[5]` Sage users can execute the following two commands to get classes MultisetPartition(s) and MultisetTableau(x) that may be helpful for experimentation. `load('http://garsia.math.yorku.ca/~zabrocki/multisetpartition.py') load('http://garsia.math.yorku.ca/~zabrocki/multisettableau.py')` [1] K. Nishiyama, an unpublished paper about the restriction problem from $Gl(n)$ to $S_n$ and its connection to plethysm. Restriction of the irreducible representations of $ GL_n $ to the symmetric group ${S}_n$ [2] R. Orellana and M. Zabrocki, Symmetric Group Characters as Symmetric Functions [3] R.C. King, Branching rules for $GL(N) \supset \Sigma_m$ and the evaluation of inner plethysm [4] This seminar from 2006-2007 and again in 2009-2010 and notes developed from that discussion. [5] T. Scharf, J. Y. Thibon, A Hopf-algebra approach to inner plethysm, Adv. in Math. 104 (1994), pp. 30--58. [6] D.E. Littlewood, Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric Groups, Canad. J. Math., {\bf 10}, 1958, 17--32. Frank: Prove conjectured Murnaghan-Nakayama for quantum Schubert polynomials click here to see more [1] A. Morrison, A Murnaghan-Nakayama rule for Schubert polynomials, FPSAC 2014 [2] A. Morrison, F. Sottile, Proof for ordinary cohomology, Annals of Combinatorics [3] K. Meszaros, G. Panova, A. Postnikov, Schur times Schubert via the Fomin-Kirillov algebra, Electronic J. Combinatorics, (2013). [4] A. Postnikov, On a quantum version of Pieri's formula, Advances in geometry, Progr. Math., vol. 172, Birkhauser Boston, Boston, MA, 1999, pp. 371--383. Alex: Understanding the "Reduce" order of the symmetric group Nantel: Complete the square for the Theta map.